Chapter 03 - Random Variables.ipynb

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   "source": [
    "## 3. Random Variables"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.1 Introduction"
   ]
  },
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   "metadata": {},
   "source": [
    "A **random variable** is a mapping $X : \\Omega \\rightarrow \\mathbb{R}$ that assigns a real number $X(\\omega)$ to each outcome $\\omega$.\n",
    "\n",
    "*Technically, a random variable must be measurable.  See the technical appendix for details.*\n",
    "\n",
    "Given a random variable $X$ and a subset $A$ of the real line, define $X^{-1}(A) = \\{ \\omega \\in \\Omega : X(\\omega) \\in A \\}$ and let\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathbb{P}(X \\in A) = \\mathbb{P}(X^{-1}(A)) &= \\mathbb{P}(\\{ \\omega \\in \\Omega; X(\\omega) \\in A \\}) \\\\\n",
    "\\mathbb{P}(X = x) = \\mathbb{P}(X^{-1}(x)) &= \\mathbb{P}(\\{ \\omega \\in \\Omega; X(\\omega) = x \\} )\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "$X$ denotes a random variable and $x$ denotes a possible value of $X$."
   ]
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   "metadata": {},
   "source": [
    "### 3.2 Distribution Functions and Probability Functions"
   ]
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   "source": [
    "The **cumulative distribution function**, CDF, $F_X : \\mathbb{R} \\rightarrow [0, 1]$ of a random variable $X$ is defined by\n",
    "\n",
    "$$ F_X(x) = \\mathbb{P}(X \\leq x) $$\n",
    "\n",
    "The following result shows that the CDF completely determines the distribution of a random variable."
   ]
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   "source": [
    "**Theorem 3.7**.  Let $X$ have CDF $F$ and let $Y$ have CDF $G$.  If $F(x) = G(x)$ for all $x$ then $\\mathbb{P}(X \\in A) = \\mathbb{P}(Y \\in A)$ for all $A$.\n",
    "\n",
    "*Technically, we only have that $\\mathbb{P}(X \\in A) = \\mathbb{P}(Y \\in A)$ for every measurable event $A$.*"
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   "source": [
    "**Theorem 3.8**.  A function $F$ mapping the real line to $[0, 1]$ is a CDF for some probability measure $\\mathbb{P}$ if and only if if satisfies the following three conditions:\n",
    "\n",
    "  - $F$ is non-decreasing, i.e. $x_1 < x_2$ implies that $F(x_1) \\leq F(x_2)$.\n",
    "  - $F$ is normalized: $\\lim_{n \\rightarrow -\\infty} F(x) = 0$ and $\\lim_{n \\rightarrow +\\infty} F(x) = 1$.\n",
    "  - $F$ is right-continuous, i.e. $F(x) = F(x^+)$ for all $x$, where\n",
    "  \n",
    "  $$ F(x^+) = \\lim_{y \\rightarrow x, y > x} F(y) $$"
   ]
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   "source": [
    "**Proof**.  Suppose that $F$ is a CDF.  Let us show that $F$ is right-continuous.  Let $x$ be a real number and let $y_1, y_2, \\dots$ be a sequence of real numbers such that $y_1 > y_2 > \\dots$ and $\\lim_i y_i = x$.  Let $A_i = (-\\infty, y_i]$ and let $A = (-\\infty, x]$.  Note that $A = \\cap_{i=1}^\\infty A_i$ and also note that $A_1 \\supset A_2 \\supset \\cdots$.  Because the events are monotone, $\\lim_i \\mathbb{P}(A_i) = \\mathbb{P}(\\cap_i A_i)$.  Thus,\n",
    "\n",
    "$$ F(x) = \\mathbb{P}(A) = \\mathbb{P}(\\cap_i A_i) = \\lim_i \\mathbb{P}(A_i) = \\lim_i F(y_i) = F(x^+) $$\n",
    "\n",
    "Showing that $F$ is non-decreasing and is normalized is similar.  Proving the other direction, that is that $F$ is non-decreasing, normalized, and right-continuous then it is a CDF for some random variable, uses some deep tools in analysis."
   ]
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   "source": [
    "$X$ is **discrete** if it takes countably many values\n",
    "\n",
    "$$ \\{ x_1, x_2, \\dots \\} $$\n",
    "\n",
    "We define the **probability density function** or **probability mass function** for $X$ by\n",
    "\n",
    "$$ f_X(x) = \\mathbb{P}(X = x) $$\n",
    "\n",
    "*A set is countable if it is finite or if it can be put in a one-to-one correspondence with the integers.*"
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    "Thus, $f_X(x) \\geq 0$ for all $x \\in \\mathbb{R}$ and $\\sum_i f_X(x_i) = 1$.  The CDF of $X$ is related to the PDF by\n",
    "\n",
    "$$ F_X(x) = \\mathbb{P}(X \\leq x) = \\sum_{x_i \\leq x} f_X(x_i) $$\n",
    "\n",
    "Sometimes we write $f_X$ and $F_X$ simply as $f$ and $F$."
   ]
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   "source": [
    "A random variable $X$ is **continuous** if there exists a function $f_X$ such that $f_X(x) \\geq 0$ for all $x$, $\\int_{-\\infty}^\\infty f_X(x) dx = 1$, and for every $a \\leq b$,\n",
    "\n",
    "$$ \\mathbb{P}(a < X < b) = \\int_a^b f_X(x) dx $$\n",
    "  \n",
    "The function $f_X$ is called the **probability density function**.  We have that\n",
    "\n",
    "$$ F_X(x) = \\int_{-\\infty}^x f_X(t) dt $$\n",
    "\n",
    "and $f_X(x) = F'_X(x)$ at all points $x$ at which $F_X$ is differentiable.\n",
    "\n",
    "Sometimes we shall write $\\int f(x) dx$ or simply $\\int f$ to mean $\\int_{-\\infty}^\\infty f(x) dx$."
   ]
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   "source": [
    "**Warning**:  Note that if $X$ is continuous then $\\mathbb{P}(X = x) = 0$ for every $x$.  We only have $f(x) = \\mathbb{P}(X = x)$ for discrete random variables; we get probabilities from a PDF by integrating."
   ]
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   "source": [
    "**Lemma 3.15**.  Let $F$ be the CDF for a random variable $X$.  Then:\n",
    "\n",
    "- $\\mathbb{P}(X = x) = F(x) - F(x^-)$ where  $F(x^-) = \\lim_{y \\uparrow x} F(y)$\n",
    "- $\\mathbb{P}(x < X \\leq y) = F(y) - F(x)$\n",
    "- $\\mathbb{P}(X > x) = 1 - F(x)$\n",
    "- If $X$ is continuous then\n",
    "\n",
    "  $$ \\mathbb{P}(a < X < b) = \\mathbb{P}(a \\leq X < b) = \\mathbb{P}(a < X \\leq b) = \\mathbb{P}(a \\leq X \\leq b) $$"
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    "It is also useful to define the inverse CDF (or quantile function).\n",
    "\n",
    "Let $X$ be a random variable with CDF $F$.  The **inverse CDF** or **quantile function** is defined by\n",
    "\n",
    "$$ F^{-1}(q) = \\inf \\{ x : F(x) \\ge q \\} $$\n",
    "\n",
    "for $q \\in [0, 1]$.  If $F$ is strictly increasing and continuous then $F^{-1}(q)$ is the unique real number $x$ such that $F(x) = q$."
   ]
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   "metadata": {},
   "source": [
    "We call $F^{-1}(1/4)$ the first quartile, $F^{-1}(1/2)$ the median (or second quartile), and $F^{-1}(3/4)$ the third quartile."
   ]
  },
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   "source": [
    "Two random variables $X$ and $Y$ are **equal in distribution** -- written $X \\overset{d}= Y$ -- if $F_X(x) = F_Y(x)$ for all $x$.  This does not mean that $X$ and $Y$ are equal.  Rather, it means that probability statements about $X$ and $Y$ will be the same."
   ]
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   "source": [
    "### 3.3 Some Important Discrete Random Variables"
   ]
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   "source": [
    "**Warning about notation**:  It is traditional to write $X \\sim F$ to indicate that $X$ has distribution $F$.  This is an unfortunate notation since the symbol $\\sim$ is also used to denote an approximation.  The notation is so pervasive we are stuck with it.\n",
    "\n",
    "*.. and we use $A \\approx B$ to denote approximation.  The LaTeX macros hint at this common usage: `\\sim` for $\\sim$, and `\\approx` for $\\approx$.*"
   ]
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   "source": [
    "#### The Point Mass Distribution\n",
    "\n",
    "$X$ has a point mass distribution at $a$, written $X \\sim \\delta_a$, if $\\mathbb{P}(X = a) = 1$, in which case\n",
    "\n",
    "$$ F(x) = \\begin{cases}\n",
    "0 &\\text{if } x < a \\\\\n",
    "1 &\\text{if } x \\geq a\n",
    "\\end{cases} $$\n",
    "\n",
    "The probability mass function is $f(x) = I(x = a)$, which takes value 1 if $x = a$ and 0 otherwise."
   ]
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   "metadata": {},
   "source": [
    "#### The Discrete Uniform Distribution\n",
    "\n",
    "Let $k > 1$ be a given integer.  Suppose that $X$ has probability mass function given by\n",
    "\n",
    "$$ f(x) = \\begin{cases}\n",
    "1/k &\\text{for } x = 1, \\dots, k \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases} $$\n",
    "\n",
    "We say that $X$ has a uniform distribution on $\\{ 1, \\dots, k \\}$."
   ]
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   "metadata": {},
   "source": [
    "#### The Bernoulli Distribution\n",
    "\n",
    "Let $X$ represent a coin flip.  Then $\\mathbb{P}(X = 1) = p$ and $\\mathbb{P}(X = 0) = 1 - p$ for some $p \\in [0, 1]$.  We say that $X$ has a Bernoulli distribution, written $X \\sim \\text{Bernoulli}(p)$.  The probability mass function is $f(x) = p^x (1 - p)^{1 - x}$ for $x \\in \\{ 0, 1 \\}$."
   ]
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   "source": [
    "#### The Binomial Distribution\n",
    "\n",
    "Suppose we have a coin which falls heads with probability $p$ for some $0 \\leq p \\leq 1$.  Flip the coin $n$ times and let $X$ be the number of heads.  Assume that the tosses are independent.  Let $f(x) = \\mathbb{P}(X = x)$ be the mass function.  It can be shown that\n",
    "\n",
    "$$ f(x) = \\begin{cases}\n",
    "\\binom{n}{x} p^x (1 - p)^{n - x} &\\text{for } x= 0, \\dots, n \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases} $$\n",
    "\n",
    "A random variable with this mass function is called a Binomial random variable, and we write $X \\sim \\text{Binomial}(n, p)$.  If $X_1 \\sim \\text{Binomial}(n, p_1)$ and $X_2 \\sim \\text{Binomial}(n, p_2)$ then $X_1 + X_2 \\sim \\text{Bimomial}(n, p_1 + p_2)$."
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that $X$ is a random variable, $x$ denotes a particular value of the random variable, and $n$ and $p$ are **parameters**, that is, fixed real numbers."
   ]
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   "source": [
    "#### The Geometric Distribution\n",
    "\n",
    "$X$ has a geometric distribution with parameter $p \\in (0, 1)$, written $X \\sim \\text{Geom}(p)$, if\n",
    "\n",
    "$$\\mathbb{P}(X = k) = p(1 - p)^{k - 1}, \\quad k \\geq 1$$\n",
    "\n",
    "We have that\n",
    "\n",
    "$$ \\sum_{k=1}^\\infty \\mathbb{P}(X = k) = p \\sum_{k=1}^\\infty (1 - p)^k = \\frac{p}{1 - (1 - p)} = 1 $$\n",
    "\n",
    "Think of $X$ as the number of flips needed until the first heads when flipping a coin."
   ]
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   "metadata": {},
   "source": [
    "#### The Poisson Distribution\n",
    "\n",
    "$X$ has a Poisson distribution with parameter $\\lambda$, written $X \\sim \\text{Poisson}(\\lambda)$, if\n",
    "\n",
    "$$ f(x) = e^{-\\lambda} \\frac{\\lambda^x}{x!}, \\quad x \\geq 0 $$\n",
    "\n",
    "Note that\n",
    "\n",
    "$$ \\sum_{x=1}^\\infty f(x) = e^{-\\lambda} \\sum_{x=1}^\\infty \\frac{\\lambda^x}{x!} = e^{-\\lambda} e^\\lambda = 1 $$\n",
    "\n",
    "The Poisson distribution is often used as a model for counts of rare events like radioactive decay and traffic accidents.  If $X_1 \\sim \\text{Poisson}(n, \\lambda_1)$ and $X_2 \\sim \\text{Poisson}(n, \\lambda_2)$ then $X_1 + X_2 \\sim \\text{Poisson}(n, \\lambda_1 + \\lambda_2)$."
   ]
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   "source": [
    "**Warning**:  We defined random variables to be mappings from a sample space $\\Omega$ to $\\mathbb{R}$ but we did not mention sample space in any of the distributions above.  Let's construct a sample space explicitly for a Bernoulli random variable.  Let $\\Omega = [0, 1]$, and define $\\mathbb{P}$ to satisfy $\\mathbb{P}([a, b]) = b - a$ for $0 \\leq a \\leq b \\leq 1$.  Fix $p \\in [0, 1]$ and define\n",
    "\n",
    "$$ X(\\omega) = \\begin{cases}\n",
    "1 &\\text{if }\\omega \\leq p \\\\\n",
    "0 &\\text{if }\\omega > p\n",
    "\\end{cases} $$\n",
    "\n",
    "Then $\\mathbb{P}(X = 1) = \\mathbb{P}(\\omega \\leq p) = \\mathbb{P}([0, p]) = p$ and $\\mathbb{P}(X = 0) = 1 - p$.  Thus, $X \\sim \\text{Bernoulli}(p)$.  We could do this for all of the distributions defined above.  In practice, we think of a random variable like a random number but formally it is a mapping defined on some sample space."
   ]
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   "source": [
    "### 3.4 Some Important Continuous Random Variables"
   ]
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   "source": [
    "#### The Uniform Distribution\n",
    "\n",
    "$X$ has a uniform distribution with parameters $a$ and $b$, written $X \\sim \\text{Uniform}(a, b)$, if\n",
    "\n",
    "$$ f(x) = \\begin{cases}\n",
    "\\frac{1}{b - a} &\\text{for } x \\in [a, b] \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases}$$\n",
    "\n",
    "where $a < b$.  The CDF is\n",
    "\n",
    "$$ F(x) = \\begin{cases}\n",
    "0 &\\text{if } x < a \\\\\n",
    "\\frac{x - a}{b - a} &\\text{if } x \\in [a, b] \\\\\n",
    "1 &\\text{if } x > b\n",
    "\\end{cases} $$"
   ]
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   "source": [
    "#### Normal (Gaussian)\n",
    "\n",
    "$X$ has a Normal (or Gaussian) distribution with parameters $\\mu$ and $\\sigma$, denoted by $X \\sim N(\\mu, \\sigma^2)$, if\n",
    "\n",
    "$$ f(x) = \\frac{1}{\\sigma \\sqrt{2 \\pi}} \\exp \\left\\{ -\\frac{1}{2} \\left( \\frac{x - \\mu}{\\sigma} \\right)^2\\right\\}, \\quad x \\in \\mathbb{R} $$\n",
    "\n",
    "where $\\mu \\in \\mathbb{R}$ and $\\sigma > 0$.  Later we shall see that $\\mu$ is the \"center\" (or mean) of the distribution and that $\\sigma$ is the \"spread\" (or standard deviation) of the distribution.  The Normal plays an important role in probability and statistics.  Many phenomena have approximately Normal distributions.  Later, we shall see that the distribution of a sum of random variables can be approximated by a Normal distribution (the central limit theorem).\n",
    "\n",
    "We say that $X$ has a **standard Normal distribution** if $\\mu = 0$ and $\\sigma = 1$.  Tradition dictates that a standard Normal random variable is denoted by $Z$.  The PDF and the CDF of a standard Normal are denoted by $\\phi(z)$ and $\\Phi(z)$.  The PDF is plotted below."
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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "xx = np.arange(-5, 5, step=0.01)\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "plt.plot(xx, norm.pdf(xx, loc=0, scale=1))\n",
    "plt.title('Standard Normal PDF')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is no closed-form expression for $\\Phi$.  Here are some useful facts:\n",
    "\n",
    "- If $X \\sim N(\\mu, \\sigma^2)$ then $Z = (X - \\mu) / \\sigma \\sim N(0, 1)$\n",
    "- If $Z \\sim N(0, 1)$ then $X = \\mu + \\sigma Z \\sim N(\\mu, \\sigma^2)$\n",
    "- If $X_i \\sim N(\\mu_i, \\sigma_i^2)$, $i = 1, \\dots, n$ are independent then\n",
    "\n",
    "  $$ \\sum_{i=1}^n X_i \\sim N \\left(\\sum_{i=1}^n \\mu_u, \\sum_{i=1}^n \\sigma_i^2 \\right) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It follows from the first fact that if $X \\sim N(\\mu, \\sigma^2)$ then\n",
    "\n",
    "$$ \\mathbb{P}(a < X < b) = \\mathbb{P}\\left( \\frac{a - \\mu}{\\sigma} < Z < \\frac{b - \\mu}{\\sigma} \\right) \n",
    "= \\Phi \\left( \\frac{b - \\mu}{\\sigma} \\right) - \\Phi \\left( \\frac{a - \\mu}{\\sigma} \\right) $$\n",
    "\n",
    "Thus we can compute any probabilities we want as long as we can compute the CDF $\\Phi(z)$ of the standard Normal.  All statistical and computing packages will compute $\\Phi(z)$ and $\\Phi^{-1}(q)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exponential Distribution\n",
    "\n",
    "$X$ has an exponential distribution with parameter $\\beta$, denoted by $X \\sim \\text{Exp}(\\beta)$, if\n",
    "\n",
    "$$ f(x) = \\frac{1}{\\beta} e^{-x / \\beta}, \\quad x > 0 $$\n",
    "\n",
    "where $\\beta > 0$.  The exponential distribution is used to model the lifetimes of electronic components and the waiting times between rare events."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Gamma Distribution\n",
    "\n",
    "For $\\alpha > 0$, the **Gamma function** is defined by $\\Gamma(\\alpha) = \\int_0^\\infty y^{\\alpha - 1} e^{-y} dy$.  $X$ has a Gamma distribution with parameters $\\alpha$ and $\\beta$, denoted by $X \\sim \\text{Gamma}(\\alpha, \\beta)$, if\n",
    "\n",
    "$$ f(x) = \\frac{1}{\\beta^\\alpha \\Gamma(\\alpha)} x^{\\alpha - 1} e^{-x / \\beta}, \\quad x > 0 $$\n",
    "\n",
    "where $\\alpha, \\beta > 0$.  The exponential distribution is just a $\\text{Gamma}(1, \\beta)$ distribution.  If $X_i \\sim \\text{Gamma}(\\alpha_i, \\beta)$ are independent, then $\\sum_{i=1}^n X_i \\sim \\text{Gamma}\\left( \\sum_{i=1}^n \\alpha_i, \\beta \\right)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Beta Distribution\n",
    "\n",
    "$X$ has Beta distribution with parameters $\\alpha > 0$ and $\\beta > 0$, denoted by $X \\sim \\text{Beta}(\\alpha, \\beta)$, if\n",
    "\n",
    "$$ f(x) = \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha) \\Gamma(\\beta)} x^{\\alpha - 1} (1 - x)^{\\beta - 1}, \\quad 0 < x < 1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $t$ and Cauchy Distribution\n",
    "\n",
    "$X$ has a $t$ distribution with $\\nu$ degrees of freedom -- written $X \\sim t_\\nu$ -- if\n",
    "\n",
    "$$ f(x) = \\frac{1}{\\sqrt{\\nu \\pi}} \\frac{\\Gamma\\left( \\frac{\\nu + 1}{2} \\right)}{\\Gamma\\left( \\frac{\\nu}{2} \\right)} \\frac{1}{\\left(1 + \\frac{x^2}{\\nu} \\right)^{(\\nu + 1) / 2}} $$\n",
    "\n",
    "The $t$ distribution is similar to a Normal but it has thicker tails.  In fact, the Normal corresponds to a $t$ with $\\nu = \\infty$.  The Cauchy distribution is a special case of the $t$ distribution corresponding to $\\nu = 1$.  The density is\n",
    "\n",
    "$$ f(x) = \\frac{1}{\\pi (1 + x^2)} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### The $\\chi^2$ Distribution\n",
    "\n",
    "$X$ has a $\\chi^2$ distribution with $p$ degrees of freedom -- written $X \\sim \\chi_p^2$ -- if\n",
    "\n",
    "$$ f(x) = \\frac{1}{\\Gamma\\left(\\frac{p}{2}\\right) 2^{\\frac{p}{2}}} x^{\\frac{p}{2} - 1} e^{-\\frac{x}{2}}, \\quad x > 0 $$\n",
    "\n",
    "If $Z_1, Z_2, \\dots$ are independent standard Normal random variables then $\\sum_{i=1}^p Z_i^2 \\sim \\chi_p^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.5 Bivariate Distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given a pair of discrete random variables $X$ and $Y$, define the **joint mass function** by $f(x, y) = \\mathbb{P}(X = x \\text{ and } Y = y)$.  From now on, we write $\\mathbb{P}(X = x \\text{ and } Y = y)$ as $\\mathbb{P}(X = x, Y = y)$.  We write $f$ as $f_{X, Y}$ when we want to be more explicit."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the continuous case, we call a function $f(x, y)$ a PDF for the random variables $(X, Y)$ if:\n",
    "- $f(x, y) \\geq 0$ for all $(x, y)$\n",
    "- $ \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty f(x, y) \\; dx dy = 1$\n",
    "- For any set $A \\subset \\mathbb{R} \\times \\mathbb{R}$, $\\mathbb{P}((X, Y) \\in A) = \\int \\int_A f(x, y) \\; dx dy$\n",
    "\n",
    "In the continuous or discrete case, we define the joint CDF as $ F_{X, Y}(x, y) = \\mathbb{P}(X \\leq x, Y \\leq y) $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.6 Marginal Distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If $(X, Y)$ have joint distribution with mass function $f_{X, Y}$, then the **marginal mass function** for $X$ is defined by\n",
    "\n",
    "$$ f_X(x) = \\mathbb{P}(X = x) = \\sum_y \\mathbb{P}(X = x, Y = y) = \\sum_y f(x, y) $$\n",
    "\n",
    "and the **marginal mass function** for $Y$ is defined by\n",
    "\n",
    "$$ f_Y(y) = \\mathbb{P}(Y = y) = \\sum_x \\mathbb{P}(X = x, Y = y) = \\sum_x f(x, y) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For continuous random variables, the marginal densities are\n",
    "\n",
    "$$ f_X(x) = \\int f(x, y) \\; dy \\quad \\text{and} \\quad f_Y(y) = \\int f(x, y) \\; dx $$\n",
    "\n",
    "The corresponding marginal CDFs are denoted by $F_X$ and $F_Y$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.7 Independent Random Variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Two random variables $X$ and $Y$ are **independent** if, for every $A$ and $B$,\n",
    "\n",
    "$$ \\mathbb{P}(X \\in A, Y \\in B) = \\mathbb{P}(X \\in A) \\mathbb{P}(Y \\in B) $$\n",
    "\n",
    "We write $X \\text{ ⫫ } Y$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In principle, to check whether $X$ and $Y$ are independent we need to check the equation above for all subsets $A$ and $B$.  Fortunately, we have the following result which we state for continuous random variables though it is true for discrete random variables too."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 3.30**.  Let $X$ and $Y$ have joint PDF $f_{X, Y}$.  Then $X \\text{ ⫫ } Y$ if and only if $f_{X, Y}(x, y) = f_X(x) f_Y(y)$ for all values $x$ and $y$.\n",
    "\n",
    "*The statement is not rigorous because the density is defined only up to sets of measure 0.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following result is helpful for verifying independence."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 3.33**.  Suppose that the range of $X$ and $Y$ is a (possibly infinite) rectangle.  If $f(x, y) = g(x) h(y)$ for some functions $g$ and $h$ (not necessarily probability density functions) then $X$ and $Y$ are independent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.8 Conditional Distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **conditional probability mass function** is\n",
    "\n",
    "$$ f_{X | Y}(x | y) = \\mathbb{P}(X = x | Y = y) = \\frac{\\mathbb{P}(X = x, Y = y)}{\\mathbb{P}(Y = y)} = \\frac{f_{X, Y}(x, y)}{f_Y(y)} $$\n",
    "\n",
    "if $f_Y(y) > 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For continuous random variables, the **conditional probability density function** is\n",
    "\n",
    "$$ f_{X | Y}(x | y) = \\frac{f_{X, Y}(x, y)}{f_Y(y)} $$\n",
    "\n",
    "assuming that $f_Y(y) > 0$.  Then,\n",
    "\n",
    "$$ \\mathbb{P}(X \\in A | Y = y) = \\int_A f_{X | Y}(x | y) \\; dx $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*We are treading in deep water here.  When we compute $\\mathbb{P}(X \\in A | Y = y)$ in the continuous case we are conditioning on the event $\\mathbb{P}(Y = y)$ which has probability 0.  We avoid this problem by defining things in terms of the PDF.  The fact that this leads to a well-defined theory is proved in more advanced courses.  We simply take it as a definition.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.9 Multivariate Distributions and IID Samples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $X = (X_1, \\dots, X_n)$ where the $X_i$'s are random variables.  We call $X$ a **random vector**.  Let $f(x_1, \\dots, x_n)$ denote the PDF.  It is possible to define their marginals, conditionals, etc. much the same way as in the bivariate case.  We say that $X_1, \\dots, X_n$ are independent if, for every $A_1, \\dots, A_n$,\n",
    "\n",
    "$$ \\mathbb{P}(X_1 \\in A_1, \\dots, X_n \\in A_n) = \\prod_{i=1}^n \\mathbb{P}(X_i \\in A_i) $$\n",
    "\n",
    "It suffices to check that $f(x_1, \\dots, x_n) = \\prod_{i=1}^n f_{X_i}(x_i)$.  If $X_1, \\dots, X_n$ are independent and each has the same marginal distribution with density $f$, we say that $X_1, \\dots, X_n$ are IID (independent and identically distributed).  We shall write this as $X_1, \\dots, X_n \\sim f$ or, in terms of the CDF, $X_1, \\dots, X_n \\sim F$.  This means that $X_1, \\dots, X_n$ are independent draws from the same distribution.  We also call $X_1, \\dots, X_n$ a **random sample** from $F$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.10 Two Important Multivariate Distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Multinomial Distribution\n",
    "\n",
    "The multivariate version of the Binomial is called a Multinomial.  Consider drawing a ball from an urn which has balls of $k$ different colors.  Let $p = (p_1, \\dots, p_k)$ where $p_j \\geq 0$ and $\\sum_{j=1}^k p_j = 1$ and suppose that $p_j$ is the probability of drawing a ball of color $j$.  Draw $n$ times (independent draws with replacement) and let $X = (X_1, \\dots, X_k)$, where $X_j$ is the number of times that color $j$ appears.  Hence, $n = \\sum_{j=1}^k X_j$.  We say that $X$ has a Multinomial distribution with parameters $n$ and $p$, written $X \\sim \\text{Multinomial}(n, p)$.  The probability function is\n",
    "\n",
    "$$ f(x) = \\binom{n}{x_1 \\cdots x_k} p_1^{x_1} \\cdots p_k^{x_k} $$\n",
    "\n",
    "where\n",
    "\n",
    "$$ \\binom{n}{x_1 \\cdots x_k} = \\frac{n!}{x_1! \\cdots x_k!} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Lemma 3.41**.  Suppose that $X \\sim \\text{Multinomial}(n, p)$ where $X = (X_1, \\dots, X_k)$ and $p = (p_1, \\dots, p_k)$.  The marginal distribution of $X_j$ is $\\text{Binomial}(n, p_j)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Multivariate Normal\n",
    "\n",
    "The univariate Normal had two parameters, $\\mu$ and $\\sigma$.  In the multivariate vector, $\\mu$ is a vector and $\\sigma$ is replaced by a matrix $\\Sigma$.  To begin, let\n",
    "\n",
    "$$ Z = \\begin{pmatrix}\n",
    "Z_1 \\\\\n",
    "\\vdots \\\\\n",
    "Z_k\n",
    "\\end{pmatrix}$$\n",
    "\n",
    "where $Z_1, \\dots, Z_k \\sim N(0, 1)$ are independent.  The density of $Z$ is\n",
    "\n",
    "$$ f(z) = \\prod_{i=1}^k f(z_i) \n",
    "= \\frac{1}{(2 \\pi)^{k/2}} \\exp \\left\\{ -\\frac{1}{2} \\sum_{j=1}^k z_j^2 \\right\\} \n",
    "= \\frac{1}{(2 \\pi)^{k/2}} \\exp \\left\\{ -\\frac{1}{2} z^T z \\right\\} \n",
    "$$\n",
    "\n",
    "We say that $Z$ has a standard multivariate Normal distribution, written $Z \\sim N(0, I)$ where it is understood that $0$ represents a vector of $k$ zeroes and $I$ is the $k \\times k$ identity matrix.\n",
    "\n",
    "More generally, a vector $X$ has multivariate Normal distribution, denoted by $X \\sim N(\\mu, \\Sigma)$, if it has density\n",
    "\n",
    "$$ f(x; \\mu, \\Sigma) = \\frac{1}{(2 \\pi)^{k/2} \\text{det}(\\Sigma)^{1/2}} \\exp \\left\\{ - \\frac{1}{2} (x - \\mu)^T \\Sigma^{-1} (x - \\mu) \\right\\} $$\n",
    "\n",
    "where $\\text{det}(\\cdot)$ denotes the determinant of a matrix, $\\mu$ is a vector of length $k$, and $\\Sigma$ is a $k \\times k$ symmetric, positive definite matrix.  Setting $\\mu = 0$ and $\\Sigma = I$ gives back the standard Normal.\n",
    "\n",
    "*A matrix $\\Sigma$ is positive definite if, for all non-zero vectors $x$, $x^T \\Sigma x > 0$.*\n",
    "\n",
    "Since $\\Sigma$ is symmetric and positive definite, it can be shown that there exists a matrix $\\Sigma^{1/2}$ -- called the square root of $\\Sigma$ -- with the following properties:\n",
    "\n",
    "- $\\Sigma^{1/2}$ is symmetric\n",
    "- $\\Sigma = \\Sigma^{1/2} \\Sigma^{1/2}$\n",
    "- $\\Sigma^{1/2}\\Sigma^{-1/2} = \\Sigma^{-1/2}\\Sigma^{1/2} = I$, where $\\Sigma^{1/2} = \\left(\\Sigma^{1/2}\\right)^{-1}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 3.42**.  If $Z \\sim N(0, I)$ and $X = \\mu + \\Sigma^{1/2} Z$ then $X \\sim N(\\mu, \\Sigma)$.  Conversely, if $X \\sim N(\\mu, \\Sigma)$, then $\\Sigma^{-1/2}(X - \\mu) \\sim N(0, I)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose we partition a random Normal vector $X$ as $X = (X_a, X_b)$, and we similarly partition\n",
    "\n",
    "$$ \\mu = \\begin{pmatrix}\\mu_a & \\mu_b \\end{pmatrix} \n",
    "\\quad\n",
    "\\Sigma = \\begin{pmatrix}\n",
    "\\Sigma_{aa} & \\Sigma_{ab} \\\\\n",
    "\\Sigma_{ba} & \\Sigma_{bb}\n",
    "\\end{pmatrix} $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 3.43**.  Let $X \\sim N(\\mu, \\Sigma)$.  Then:\n",
    "\n",
    "1.  The marginal distribution of $X_a$ is $X_a \\sim N(\\mu_a, \\Sigma_{aa})$.\n",
    "2.  The conditional distribution of $X_b$ given $X_a = x_a$ is\n",
    "\n",
    "    $$  X_b | X_a = x_a \\sim N \\left( \\mu_b + \\Sigma_{ba}\\Sigma_{aa}^{-1}(x_a - \\mu_a), \\Sigma_{bb} - \\Sigma_{ba} \\Sigma_{aa}^{-1} \\Sigma_{ab} \\right) $$\n",
    "\n",
    "3.  If $a$ is a vector then $a^T X \\sim N(a^T \\mu, a^T \\Sigma a)$\n",
    "4.  $V = (X - \\mu)^T \\Sigma^{-1} (X - \\mu) \\sim \\chi_k^2$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.11 Transformations of Random Variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose that $X$ is a random variable with PDF $f_X$ and CDF $F_X$.  Let $Y = r(X)$ be a function of $X$, such as $Y = X^2$ or $Y = e^X$.  We call $Y = r(X)$ a transformation of $X$.  How do we compute the PDF and the CDF of $Y$?  In the discrete case, the answer is easy.  The mass function of $Y$ is given by\n",
    "\n",
    "$$ f_Y(y) = \\mathbb{P}(Y = y) = \\mathbb{P}(r(X) = y) = \\mathbb{P}(\\{x; r(x) = y\\}) = \\mathbb{P}(X \\in r^{-1}(y)) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The continuous case is harder.  There are 3 steps for finding $f_Y$:\n",
    "\n",
    "1. For each $y$, find the set $A_y = \\{ x : r(x) \\leq y \\}$.\n",
    "\n",
    "2. Find the CDF\n",
    "\n",
    "   $$ F_Y(y) = \\mathbb{P}(Y \\leq y) = \\mathbb{P}(r(X) \\leq y) = \\mathbb{P}(\\{x ; r(x) \\leq y \\}) = \\int_{A_y} f_X(x) dx $$\n",
    "\n",
    "3. The PDF is $f_Y(y) = F'_Y(y)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When $r$ is strictly monotone increasing or strictly monotone decreasing then $r$ has an inverse $s = r^{-1}$ and in this case one can show that\n",
    "\n",
    "$$ f_Y(y) = f_X(s(y)) \\;\\Bigg| \\frac{ds(y)}{dy} \\Bigg|$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.12 Transformation of Several Random Variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In some cases we are interested in the transformation of several random variables.  For example, if $X$ and $Y$ are given random variables, we might want to know the distribution of $X / Y$, $X + Y$, $\\max \\{ X, Y \\}$ or $\\min \\{ X, Y \\}$.  Let $Z = r(X, Y)$.  The steps for finding $f_Z$ are the same as before:\n",
    "\n",
    "1. For each $z$, find the set $A_z = \\{ (x, y) : r(x, y) \\leq z \\}$.\n",
    "2. Find the CDF\n",
    "\n",
    "  $$ F_Z(z) = \\mathbb{P}(Z \\leq z) = \\mathbb{P}(r(X, Y) \\leq z) = \\mathbb{P}(\\{ (x, y) : r(x, y) \\leq z \\})\n",
    "  = \\int \\int_{A_z} f_{X, Y}(x, y) \\; dx dy \n",
    "  $$\n",
    "  \n",
    "3. Find $f_Z(z) = F'_Z(z)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.13 Technical Appendix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall that a probability measure $\\mathbb{P}$ is defined on a $\\sigma$-field $\\mathcal{A}$ of a sample space $\\Omega$.  A random variable $X$ is **measurable** map $X : \\Omega \\rightarrow \\mathbb{R}$.  Measurable means that, for every $x$, $\\{ \\omega : X(\\omega) \\leq x \\} \\in \\mathcal{A} $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.14 Exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.1**.  Show that\n",
    "\n",
    "$$ \\mathbb{P}(X = x) = F(x^+) - F(x^-) $$\n",
    "\n",
    "and\n",
    "\n",
    "$$ F(x_2) - F(x_1) = \\mathbb{P}(X \\leq x_2) - \\mathbb{P}(X \\leq x_1) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "By definition,  $F(x_2) = \\mathbb{P}(X \\leq x_2)$ and $F(x_1) = \\mathbb{P}(X \\leq x_1)$, so the second equation is immediate.\n",
    "\n",
    "Also by definition,\n",
    "\n",
    "$$ F(x^+) = \\lim_{y \\rightarrow x, y > x} F(y) = \\lim_{y \\rightarrow x, y > x} \\mathbb{P}(X \\leq y) \n",
    "= \\mathbb{P}(\\exists y : X \\leq y, y > x) \n",
    "= \\mathbb{P}(X \\leq x) $$\n",
    "\n",
    "and\n",
    "\n",
    "$$ F(x^-) = \\lim_{y \\rightarrow x, y < x} F(y) = \\lim_{y \\rightarrow x, y > x} \\mathbb{P}(X \\leq y) \n",
    "= \\mathbb{P}(\\exists y : X \\leq y, y < x)\n",
    "= \\mathbb{P}(X < x)\n",
    "$$\n",
    "\n",
    "and so\n",
    "\n",
    "$$ F(x^+) - F(x^-) = \\mathbb{P}(X \\leq x) - \\mathbb{P}(X < x) = \\mathbb{P}(X(\\omega) \\in \\{x : X(\\omega) \\leq x \\} - \\{x: X(\\omega) < x \\}) = \\mathbb{P}(X = x)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.2**.  Let $X$ be such that $\\mathbb{P}(X = 2) = \\mathbb{P}(X = 3) = 1/10$ and $\\mathbb{P}(X = 5) = 8/10$.  Plot the CDF $F$.  Use $F$ to find $\\mathbb{P}(2 < X \\leq 4.8)$ and $\\mathbb{P}(2 \\leq X \\leq 4.8)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "\n",
    "# Draw horizontal lines\n",
    "for xmin, xmax, y in [(0, 2, 0), (2, 3, 1/10), (3, 5, 2/10), (5, 10, 1)]:\n",
    "    plt.hlines(y, xmin, xmax, color='C0', linestyle='solid')\n",
    "    \n",
    "# Draw vertical lines\n",
    "for ymin, ymax, x in [(0, 1/10, 2), (1/10, 2/10, 3), (2/10, 1, 5)]:\n",
    "    plt.vlines(x, ymin, ymax, color='C0', linestyle='dashed')\n",
    "    \n",
    "# Mark open intervals\n",
    "plt.scatter([2, 3, 5], [0, 1/10, 2/10], color='C0', facecolor='white', zorder=10, linewidth=2)\n",
    "\n",
    "# Mark close intervals\n",
    "plt.scatter([2, 3, 5], [1/10, 2/10, 1], color='C0', facecolor='C0', zorder=10, linewidth=2)\n",
    "\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('F(x)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\mathbb{P}(2 < X \\leq 4.8) = F(4.8) - F(2) = 0.2 - 0.1 = 0.1\n",
    "\\quad \\text{ and } \\quad\n",
    "\\mathbb{P}(2 \\leq X \\leq 4.8) = F(4.8) - F(2^{-}) = 0.2 - 0 = 0.2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.3**.  Prove Lemma 3.15.\n",
    "\n",
    "Let $F$ be the CDF for a random variable $X$.  Then:\n",
    "\n",
    "- $\\mathbb{P}(X = x) = F(x) - F(x^-)$ where  $F(x^-) = \\lim_{y \\uparrow x} F(y)$\n",
    "- $\\mathbb{P}(x < X \\leq y) = F(y) - F(x)$\n",
    "- $\\mathbb{P}(X > x) = 1 - F(x)$\n",
    "- If $X$ is continuous then\n",
    "\n",
    "  $$ \\mathbb{P}(a < X < b) = \\mathbb{P}(a \\leq X < b) = \\mathbb{P}(a < X \\leq b) = \\mathbb{P}(a \\leq X \\leq b) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "Proof of first statement:\n",
    "\n",
    "$$ F(x) - F(x^{-}) = \\mathbb{P}(X \\leq x) - \\mathbb{P}(X < x) = \\mathbb{P}(\\{\\omega : X(\\omega) \\leq x\\} - \\{\\omega : X(\\omega) < x\\}) = \\mathbb{P}(X = x)$$\n",
    "\n",
    "Proof of second statement:\n",
    "\n",
    "$$ \\mathbb{P}(x < X \\leq y) = \\mathbb{P}(\\{\\omega : X(\\omega) \\leq y\\} - \\{\\omega : X(\\omega) \\leq x\\}) = \\mathbb{P}(X \\leq y) - \\mathbb{P}(X \\leq x) = F(y) - F(x) $$\n",
    "\n",
    "Proof of third statement:\n",
    "\n",
    "$$ \\mathbb{P}(X > x) = \\mathbb{P}(\\{\\omega: X(\\omega) > x\\}) = \\mathbb{P}(\\{\\omega: X(\\omega) \\leq x\\}^c) = 1 - \\mathbb{P}(X \\leq x) = 1 - F(x) $$\n",
    "\n",
    "Proof of fourth statement:\n",
    "\n",
    "If $F$ is continuous, then $\\mathbb{P}(X = a) = \\mathbb{P}(X = b) = 0$, and so\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathbb{P}(a < X \\leq b) &= \\mathbb{P}(a < X < b) + \\mathbb{P}(X = b) = \\mathbb{P}(a < X < b) \\\\\n",
    "\\mathbb{P}(a \\leq X < b) &= \\mathbb{P}(X = a) + \\mathbb{P}(a < X < b) = \\mathbb{P}(a < X < b) \\\\\n",
    "\\mathbb{P}(a \\leq X \\leq b) &= \\mathbb{P}(X = a) + \\mathbb{P}(a < X < b) + \\mathbb{P}(X = b) = \\mathbb{P}(a < X < b)\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.4**.  Let $X$ have probability density function\n",
    "\n",
    "$$ f_X(x) = \\begin{cases}\n",
    "1/4 &\\text{if } 0 < x < 1 \\\\\n",
    "3/8 &\\text{if } 3 < x < 5 \\\\\n",
    "0   &\\text{otherwise}\n",
    "\\end{cases} $$\n",
    "\n",
    "**(a)**  Find the cumulative distribution function of $X$.\n",
    "\n",
    "**(b)**  Let $Y = 1 / X$.  Find the probability density function $f_Y(y)$ for $Y$.  Hint:  Consider three cases, $\\frac{1}{5} \\leq y \\leq \\frac{1}{3}$, $\\frac{1}{3} \\leq y \\leq 1$, and $y \\geq 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "**(a)**\n",
    "\n",
    "$$ \n",
    "F_X(x) = \\begin{cases}\n",
    "0 &\\text{if } x \\leq 0 \\\\\n",
    "\\frac{x}{4} &\\text{if } 0 < x \\leq 1 \\\\\n",
    "\\frac{1}{4} &\\text{if } 1 < x \\leq 3 \\\\\n",
    "\\frac{1}{4} + \\frac{3(x - 3)}{8} &\\text{if } 3 < x \\leq 5 \\\\\n",
    "1 &\\text{if } x \\geq 5\n",
    "\\end{cases} \n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "\n",
    "# Draw horizontal lines\n",
    "for xmin, xmax, y in [(-2, 0, 0), (1, 3, 1/4), (5, 7, 1)]:\n",
    "    plt.hlines(y, xmin, xmax, color='C0', linestyle='solid')\n",
    "    \n",
    "# Draw diagonal lines\n",
    "for x1, y1, x2, y2 in [(0, 0, 1, 1/4), (3, 1/4, 5, 1)]:\n",
    "    plt.plot([x1, x2], [y1, y2], color='C0')\n",
    "    \n",
    "plt.xlabel(r'$x$')\n",
    "plt.ylabel(r'$F(x)$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(b)**\n",
    "\n",
    "$\\mathbb{P}(X > 0) = 1$, so $\\mathbb{P}(Y > 0) = 1$.  For $y > 0$, we have:\n",
    "\n",
    "$$ F_Y(y) = \\mathbb{P}(Y \\leq y) = \\mathbb{P}\\left(\\frac{1}{X} \\leq y\\right) = \\mathbb{P}\\left(X \\geq \\frac{1}{y}\\right) = 1 - F_X\\left(\\frac{1}{y} \\right) $$\n",
    "\n",
    "so\n",
    "\n",
    "\n",
    "$$ F_Y(y) = \\begin{cases}\n",
    "0 &\\text{if } y \\leq \\frac{1}{5} \\\\\n",
    "\\frac{15}{8} - \\frac{3}{8y} &\\text{if } \\frac{1}{5} < y \\leq \\frac{1}{3} \\\\\n",
    "\\frac{3}{4} &\\text{if } \\frac{1}{3} < y \\leq 1 \\\\\n",
    "1 - \\frac{1}{4y} &\\text{if } y > 1\n",
    "\\end{cases} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ZP+OvKgBWL10KWBVqrfnQF2/N/3f5DVk/M5n/+qYd+aHzThx1WQDwuARqYOT2zLXyLz729XzmunvzA+dsza/+2AU5YePsqMsCgCMiUAMjdf1de/LTv3dV7nhgf979iqfnp/63M+3gAcBRRaAGRuZjV96ed//xtdmybiofueT52XHGsaMuCQCeMIEaGLp2p5t/+6fX58NfujUvOOu4/Obrnp2tG2dGXRYAPCkCNTBUu/e38rbf/0q+8M37c8n3n5Vf+pFzMjmxZtRlAcCTJlADQ3Prd/fmLR+8Irft2pf/8H9ckFdfuH3UJQHAignUwFB89bYH8pYPXpGa5MM/9bw8/6zjRl0SAPSFQA0M3Be+uTP//MNX5fgNM/nQWy7KGcevH3VJANA3AjUwUJ+6+q78wke/lqeesDG/85YL7S8NwNgRqIGB+b2/vzXv/uNrc+Hpx+a//sSObF47NeqSAKDvBGpgID78xe/kV/7kurzk3BPyvjc8J7NTE6MuCQAGYqh7VZVSLi6l3FhKubmU8s7DvL+5lPKnpZSvl1KuK6W8eZj1Af3xu1+6Nb/yJ9flB59+Qi5943OFaQDG2tACdSllIsl7k7wsyXlJXldKOe+Q034myfW11mcmeXGS/1RKmR5WjcDKLY15vPTcE/LeNzwn05P2mAZgvA2z012U5OZa6y211oUkH0nyykPOqUk2llJKkg1JdiVpD7FGYAX+x5W355f/6NremMcbn5OZSSvTAIy/YQbqbUluX/b6jsVjy/1WkqcnuSvJNUl+rtbaHU55wEr85Q335l2fuCYvOvv4vF+YBqBBhhmoy2GO1UNe/0iSryU5JcmzkvxWKWXTo36jUi4ppVxZSrly586d/a8UeEKuunVXfub3v5JnnLIpl77xucI0AI0yzEB9R5Ll9xk+Nb2V6OXenOQTtefmJN9Ocu6hv1Gt9bJa645a646tW7cOrGDg8X3z3ofylg9emZM2zeYDP3lh1s/YPAiAZhlmoL4iydmllDMXLzR8bZJPHnLObUlemiSllBOTnJPkliHWCDwB9+2Zy0984MuZmliTD73leTl+w8yoSwKAoRvaUlKttV1KeVuSzySZSPKBWut1pZS3Lr5/aZJ/l+SDpZRr0hsReUet9f5h1QgcublWJ//sw1flgX2tfOytL8hpx60bdUkAMBJD/bfZWuvlSS4/5Nily57fleSHh1kT8MTVWvOuT1yTr9/+YC5943Nz/rbNoy4JAEbGBrHAE/Zf/vaW/NFX78z/9UNPy8XnnzTqcgBgpARq4An525t25lc//Y284oKT8/aXPHXU5QDAyAnUwBG7Z/dcfv6jX8vZJ2zIr/3YBendgwkAmk2gBo5Iu9PN2//gK5lrdfK+Nzwn66ZtjwcAyZAvSgSOXv/xszfliu88kN94zbPy1BM2jrocAFg1rFADj+sL39yZS//mW3ndRdvzqmdvG3U5ALCqCNTA97R7Xyv/4mNX56knbMi//kfPGHU5ALDqCNTA9/Qrf3Jt7n94Pu959bMyOzUx6nIAYNURqIHH9Kdfvyuf/Ppd+dmXnp3vO9XNWwDgcARq4LDu2zOXX/mTa/PM7Vvy0y9+yqjLAYBVS6AGDuvf/un12bfQya+/+pmZnPBXBQA8Fl0SeJS/+sa9+bNr7s7PvuSpecrWDaMuBwBWNYEaeIR9C+38yh9fl6eesCGXfL9RDwB4PG7sAjzCe/7iptz54P587K0vyPSk/+YGgMcjUMMQferqu3Lp57+V9TOr86u3d76dG+55KK+7aHsuPOPYUZcDAEcFy08wRL/4P76e7+5dGHUZj6kmee7px+QdF5876lIA4KixOpfJYAx1uzXz7W5evWN7fuGHnjbqcgCAPrFCDUOyv9VJkqybdrdBABgnAjUMyVKgXitQA8BYEahhSPYvLAbqKYEaAMaJQA1DYoUaAMaTQA1Dsm/BDDUAjCOBGobk4MiHzXUAYJwI1DAk+1vtJEY+AGDcCNQwJPsXukmMfADAuBGoYUj2LSyuUNvlAwDGikANQzJnlw8AGEsCNQzJPvtQA8BYEqhhSA7sQy1QA8BYEahhSPYvdDIzuSZr1pRRlwIA9JFADUOyv9WxwwcAjCGBGoZk30LHuAcAjCGBGoZkf6tjhw8AGEMCNQzJ/gWBGgDGkUANQ7J/oZN1U5OjLgMA6DOBGoZkX6uTWSvUADB2BGoYkrmFTta5KBEAxo5ADUOyr9U2Qw0AY0ighiHZv9AVqAFgDAnUMCT7F9pGPgBgDAnUMAS1VvtQA8CYEqhhCObb3XRrBGoAGEMCNQzBXKuTJG49DgBjSKCGIdi30AvU66xQA8DYEahhCPYvrlDPWqEGgLEjUMMQ7D+wQu3W4wAwbgRqGIL9ZqgBYGwJ1DAESzPUdvkAgPEjUMMQLI18WKEGgPEjUMMQ7G+1k1ihBoBxJFDDEMy1ukmsUAPAOBKoYQjmDmyb5ysHAONGd4chWFqhtg81AIwfgRqGYGmFembSVw4Axo3uDkMw1+5kZnJNSimjLgUA6DOBGoZgvtU17gEAY0qghiHYv9BxQSIAjCkdHoZgrt2xQg0AY0qghiGYa3UyOylQA8A4EqhhCOZaXSMfADCmdHgYgrlWJzNGPgBgLAnUMARzbbt8AMC4EqhhCOZbncy6qQsAjCUdHoZgrmWXDwAYVwI1DIGLEgFgfOnwMAT2oQaA8SVQwxAY+QCA8SVQw4DVWnsjHy5KBICxpMPDgM23u0liH2oAGFMCNQzYfKsXqI18AMB4EqhhwObanSSxywcAjCkdHgZsrrUYqCetUAPAOBpqoC6lXFxKubGUcnMp5Z2Pcc6LSylfK6VcV0r5m2HWB4MwZ+QDAMba5LD+oFLKRJL3JvmhJHckuaKU8sla6/XLztmS5H1JLq613lZKOWFY9cGgHFihNvIBAGNpmB3+oiQ311pvqbUuJPlIklcecs7rk3yi1npbktRa7xtifTAQBwO1FWoAGEfDDNTbkty+7PUdi8eWe1qSY0opny+lXFVKedPQqoMB2W+FGgDG2tBGPpKUwxyrh7yeTPLcJC9NsjbJF0spX6q13vSI36iUS5JckiSnnXbaAEqF/lmaoZ5xUSIAjKVhLpndkWT7stenJrnrMOd8uta6t9Z6f5K/TfLMQ3+jWutltdYdtdYdW7duHVjB0A/zi9vmrZ0WqAFgHA0zUF+R5OxSypmllOkkr03yyUPO+ZMkLyqlTJZS1iV5XpIbhlgj9J0ZagAYb0Mb+ai1tkspb0vymSQTST5Qa72ulPLWxfcvrbXeUEr5dJKrk3ST/Hat9dph1QiDcGDbvEkz1AAwjoY5Q51a6+VJLj/k2KWHvP61JL82zLpgkKxQA8B4s2QGA+bGLgAw3gRqGLC5didTEyUTaw630Q0AcLQTqGHA5lqdzNoyDwDGlkANAzbX6mbGuAcAjC2BGgZsvtVxl0QAGGO6PAzYXLvjgkQAGGMCNQzYXKtrhRoAxpguDwPmokQAGG8CNQzYXMvIBwCMM4EaBszIBwCMN10eBmy+3bFtHgCMMYEaBmy+3c3MhK8aAIwrXR4GbKHdzYyRDwAYW7o8DNh8u5sZu3wAwNgSqGHA5tudTE/6qgHAuNLlYYBqrb2RD4EaAMaWLg8D1O7WdGsEagAYY7o8DNB8u5skRj4AYIzp8jBAC4uB2kWJADC+BGoYoPl2J4mRDwAYZ7o8DNB8y8gHAIw7XR4GaKFj5AMAxp1ADQO0tEJt5AMAxpcuDwO0NENt5AMAxpcuDwN0cJcPXzUAGFe6PAzQ0j7UM1NmqAFgXAnUMEAHRj4mfNUAYFzp8jBAB1eofdUAYFzp8jBA82aoAWDs6fIwQEuB2i4fADC+dHkYoIO7fLgoEQDG1RMO1KWU9aUU6QCOwNJFiUY+AGB8PW6XL6WsKaW8vpTyZ6WU+5J8I8ndpZTrSim/Vko5e/BlwtFpaYXaLh8AML6OpMv/dZKnJHlXkpNqrdtrrSckeVGSLyX596WUNw6wRjhqzbe7mZ5YkzVryqhLAQAGZPIIzvnBWmvr0IO11l1JPp7k46WUqb5XBmNgvtU17gEAY+5xO/1SmC6lfK6U8szvdQ7wSAudjh0+AGDMPZFO/0tJ3lNK+e+llJMHVRCMEyvUADD+jrjT11q/Umt9SZJPJfl0KeVfl1LWDq40OPrNt7uZmbIpDgCMsye0dFZKKUluTPL+JG9P8s1Syo8PojAYBwuLFyUCAOPriDt9KeXvktyZ5D1JtiX5ySQvTnJRKeWyQRQHR7v5diczUwI1AIyzI9nlY8lbk1xXa62HHH97KeWGPtYEY2O+bYYaAMbdkdzYpSRJrfXaw4TpJa/oa1UwJhbaXbt8AMCYO6Ibu5RS3l5KOW35wVLKdCnlJaWU30ny/YMpD45uvRVqFyUCwDg7kpGPi5O8JckflFLOTPJgktkkE0k+m+Q9tdavDa5EOHrNtztGPgBgzD1uoK61ziV5X5L3Ld4R8fgk+2utDw66ODjaGfkAgPH3hDr94h0RnyNMw5FxUSIAjL8n0+m/r5Tyz/peCYwhM9QAMP6ecKCutf77JK1SyntKKS8qpWwYQF0wFox8AMD4e8KdvpTyfyZ5VZJrk1yU3l0TgcNwUSIAjL/HvSixlPI7tdafWHbonlrrqwZYE4yFbrem1alGPgBgzB3JtnkXLD0ppXy21vrDA6wHxsZCp5skRj4AYMwdSadffnfErYMqBMbNfKsXqI18AMB4O5IV6pNKKT+Z5OtJymDLgfEx3+kkSWamBGoAGGdHEqj/TZIdSd6c5NRSyjVJrlv8ub7W+vHBlQdHr6UV6ukJgRoAxtmR3CnxsuWvSymnpjdX/X3p7fYhUMNhzLcXRz6mXJQIAOPsSFaoH6HWekeSO5Jc3v9yYHwstM1QA0AT6PQwIPPt3gy1XT4AYLzp9DAgB1aozVADwFjT6WFAWp3ejpNWqAFgvOn0MCALi9vmTVmhBoCxptPDgCy0eyvUAjUAjDedHgbErccBoBl0ehiQVtuNXQCgCXR6GBAr1ADQDDo9DEhrMVBPTZQRVwIADJJADQOytA+1FWoAGG86PQzIwoEVal8zABhnOj0MyIKLEgGgEXR6GJBWp5vJNSVr1pihBoBxJlDDgCy0u+anAaABdHsYkFanmp8GgAbQ7WFA5q1QA0Aj6PYwIK1O1wWJANAAQ+32pZSLSyk3llJuLqW883ucd2EppVNK+bFh1gf91Op03dQFABpgaIG6lDKR5L1JXpbkvCSvK6Wc9xjn/WqSzwyrNhgEFyUCQDMMs9tflOTmWusttdaFJB9J8srDnPf2JB9Pct8Qa4O+661QC9QAMO6G2e23Jbl92es7Fo8dUErZluSfJLl0iHXBQLgoEQCaYZjd/nDDpPWQ17+R5B211s73/I1KuaSUcmUp5cqdO3f2rUDoJyvUANAMk0P8s+5Isn3Z61OT3HXIOTuSfKSUkiTHJ3l5KaVda/3j5SfVWi9LclmS7Nix49BQDqvCQrub9TPD/IoBAKMwzG5/RZKzSylnJrkzyWuTvH75CbXWM5eel1I+mORTh4ZpOFq4sQsANMPQAnWttV1KeVt6u3dMJPlArfW6UspbF983N81YWWjbhxoAmmCo/x5da708yeWHHDtskK61/uQwaoJBaXW6mXJRIgCMPd0eBmTeCjUANIJuDwPS6nQzPelOiQAw7gRqGJAF2+YBQCPo9jAgLSMfANAIuj0MyIKLEgGgEXR7GIBaa1qdaoUaABpAt4cBWOh0kyTTVqgBYOzp9jAArU5NEivUANAAuj0MwEK7t0I9NWHbPAAYdwI1DEDrwMjHxIgrAQAGTaCGAbBCDQDNIVDDALgoEQCaQ7eHATgw8uGiRAAYe7o9DMDBkQ9fMQAYd7o9DEDLyAcANIZuDwMwb4UaABpDt4cBOHBjFyvUADD2dHsYgKUZahclAsD40+1hAMxQA0Bz6PYwAG7sAgDNIVDDALixCwA0h24PA2CGGgCaQ7eHATBDDQDNodvDALhTIgA0h24PA2CFGgCaQ7eHAVhaoZ5cY5cPABh3AjUMwEKnZnpiTUoRqAFg3AnUMAAL7a49qAGgIQRqGIB2t2t+GgAaQseHAWh1upm0wwcANIKOD/S+Xq4AAA1dSURBVAPQ6tRMuSARABpBoIYBaHW6mTLyAQCNoOPDALQ71ZZ5ANAQAjUMQKvTdZdEAGgIHR8GQKAGgObQ8WEA2t2aSftQA0AjCNQwAL0bu/h6AUAT6PgwAO1udadEAGgIgRoGoG2GGgAaQ8eHAVjo1Eyu8fUCgCbQ8WEAeivURj4AoAkEahgA2+YBQHPo+DAArY5t8wCgKQRqGIB2t5tpK9QA0Ag6PgyAFWoAaA6BGgag1ena5QMAGkLHhwFodbqZnvT1AoAm0PFhANqdmsk1Rj4AoAkEauizWuvircd9vQCgCXR86LNWpyaJG7sAQEMI1NBn7W43STJphRoAGkHHhz47uELt6wUATaDjQ5+1Or0VaiMfANAMAjX0WXtxhdo+1ADQDDo+9JkVagBoFoEa+uxgoPb1AoAm0PGhz9pdFyUCQJPo+NBnC+2lbfOMfABAEwjU0GcHV6gFagBoAoEa+swMNQA0i44PfbYUqG2bBwDNoONDny3tQz09aeQDAJpAoIY+s0INAM2i40OftZbulOiiRABoBIEa+mxphXraRYkA0Ag6PvRZu7u0D7WvFwA0gY4PfbY08mEfagBoBoEa+sw+1ADQLDo+9NnStnmTa6xQA0ATCNTQZwdWqCd9vQCgCXR86LMDM9T2oQaARtDxoc/aB2aojXwAQBMI1NBnSyMfE2aoAaARhhqoSykXl1JuLKXcXEp552Hef0Mp5erFn/9VSnnmMOuDfmh1a6YmSkoRqAGgCYYWqEspE0nem+RlSc5L8rpSynmHnPbtJP+w1npBkn+X5LJh1Qf90mp3bZkHAA0yzK5/UZKba6231FoXknwkySuXn1Br/V+11gcWX34pyalDrA/6ot2ttswDgAYZZqDeluT2Za/vWDz2WH4qyZ8f7o1SyiWllCtLKVfu3LmzjyXCyrU63UzbMg8AGmOYXf9wS3b1sCeW8gPpBep3HO79WutltdYdtdYdW7du7WOJsHKtTjeTtswDgMaYHOKfdUeS7cten5rkrkNPKqVckOS3k7ys1vrdIdUGfdPu1EzaMg8AGmOYy2hXJDm7lHJmKWU6yWuTfHL5CaWU05J8IsmP11pvGmJt0DcLnW6mXZQIAI0xtBXqWmu7lPK2JJ9JMpHkA7XW60opb118/9Ik/yrJcUnet7jlWLvWumNYNUI/WKEGgGYZ5shHaq2XJ7n8kGOXLnv+T5P802HWBP3W7to2DwCaRNeHPlvo1EwK1ADQGLo+9Fm7082UfagBoDEEauizdqca+QCABtH1oc8WOl0XJQJAgwjU0Gftrm3zAKBJdH3os1bbtnkA0CQCNfRZq9u1ywcANIiuD33W6Va7fABAgwjU0GftTs3EGl8tAGgKXR/6rHenRCvUANAUAjX0WW+FWqAGgKYQqKHP2l03dgGAJtH1oc/ana4VagBoEIEa+qzdtQ81ADSJQA191u7WTFqhBoDGEKihj2qt6XRrJm2bBwCNoetDH7W7NUmsUANAgwjU0EedpUBtlw8AaAxdH/qo1ekmsUINAE0iUEMfHVyhFqgBoCkEauijVscMNQA0jUANfWSGGgCaR9eHPlqaoXanRABoDoEa+mhphXrKDDUANIZADX3U7i6tUPtqAUBT6PrQR0s3dpky8gEAjSFQQx+1F3f5MEMNAM0hUEMfHVihtssHADSGrg991LbLBwA0jkANfdR2p0QAaByBGvqofeBOib5aANAUuj700dK2eVaoAaA5BGroo4Mr1AI1ADSFQA19dGCG2sgHADSGrg99ZOQDAJpHoIY+6nSNfABA0wjU0Ectu3wAQOPo+tBHHSMfANA4AjX0UcsuHwDQOAI19NGBGeoJXy0AaApdH/qo1emNfExYoQaAxhCooY+WVqinzFADQGMI1NBHSzd2sUINAM0hUEMfLd16fMq2eQDQGLo+9FG7200pyRor1ADQGAI19FG7W61OA0DD6PzQR+1O1/w0ADSMQA191O5Wd0kEgIYRqKGP2p3qLokA0DACNfRRb4Xa1woAmkTnhz5qd7pWqAGgYQRq6KOOGWoAaByBGvqo1a2ZtG0eADSKzg991Oka+QCAphGooY9anWofagBoGIEa+qjTrZmyywcANIrOD33UcqdEAGgcgRr6qLdCLVADQJMI1NBHbTPUANA4AjX0UbvbNUMNAA2j80MftbtWqAGgaQRq6KN2x41dAKBpdH7oo7YbuwBA4wjU0Eftbs2EXT4AoFEEauijdqdmygo1ADSKQA191OnWTJihBoBG0fmhj1qdrhu7AEDDCNTQRx3b5gFA4wjU0Ee9FWpfKwBoEp0f+sgKNQA0j0ANfdTq1kyaoQaARhGooY863erGLgDQMEMN1KWUi0spN5ZSbi6lvPMw75dSym8uvn91KeU5w6wPVqLWuhio/XcqADTJ0Dp/KWUiyXuTvCzJeUleV0o575DTXpbk7MWfS5K8f1j1wUq1uzVJrFADQMMMcyntoiQ311pvqbUuJPlIklcecs4rk3yo9nwpyZZSyslDrBGetHZnMVDb5QMAGmVyiH/WtiS3L3t9R5LnHcE525LcPdjSnpzX/JcvjroEVhEr1ADQTMNcSjtcyqhP4pyUUi4ppVxZSrly586dfSkO+mH9zERO3Dw76jIAgCEa5gr1HUm2L3t9apK7nsQ5qbVeluSyJNmxY8ejAvewfPSfv2BUfzQAAKvEMFeor0hydinlzFLKdJLXJvnkIed8MsmbFnf7eH6S3bXWVTnuAQAAyRBXqGut7VLK25J8JslEkg/UWq8rpbx18f1Lk1ye5OVJbk6yL8mbh1UfAAA8GcMc+Uit9fL0QvPyY5cue16T/MwwawIAgJWwvxcAAKyAQA0AACsgUAMAwAoI1AAAsAICNQAArIBADQAAKyBQAwDACgjUAACwAgI1AACsgEANAAArIFADAMAKCNQAALACAjUAAKyAQA0AACsgUAMAwAqUWuuoa1iRUsrOJLeO6I8/Psn9I/qzGR6f8/jzGTeDz7kZfM7NMKrP+fRa69ZDDx71gXqUSilX1lp3jLoOBsvnPP58xs3gc24Gn3MzrLbP2cgHAACsgEANAAArIFCvzGWjLoCh8DmPP59xM/icm8Hn3Ayr6nM2Qw0AACtghRoAAFZAoF6BUsqvlVK+UUq5upTyR6WULaOuif4ppVxcSrmxlHJzKeWdo66H/iulbC+l/HUp5YZSynWllJ8bdU0MRillopTy1VLKp0ZdC4NTStlSSvnDxd58QynlBaOuif4qpfzC4t/X15ZS/qCUMjvqmhKBeqX+Isn5tdYLktyU5F0jroc+KaVMJHlvkpclOS/J60op5422KgagneQXa61PT/L8JD/jcx5bP5fkhlEXwcD95ySfrrWem+SZ8ZmPlVLKtiQ/m2RHrfX8JBNJXjvaqnoE6hWotX621tpefPmlJKeOsh766qIkN9dab6m1LiT5SJJXjrgm+qzWenet9SuLzx9Kr/luG21V9Fsp5dQkr0jy26OuhcEppWxK8v1J/luS1FoXaq0PjrYqBmAyydpSymSSdUnuGnE9SQTqfnpLkj8fdRH0zbYkty97fUcErbFWSjkjybOT/P1oK2EAfiPJLyXpjroQBuqsJDuT/PfF8Z7fLqWsH3VR9E+t9c4k/zHJbUnuTrK71vrZ0VbVI1A/jlLK5xbndA79eeWyc345vX86/r3RVUqflcMcsyXOmCqlbEjy8SQ/X2vdM+p66J9Syo8mua/WetWoa2HgJpM8J8n7a63PTrI3ietfxkgp5Zj0/rX4zCSnJFlfSnnjaKvqmRx1AatdrfUHv9f7pZSfSPKjSV5a7UE4Tu5Isn3Z61OzSv5Zif4qpUylF6Z/r9b6iVHXQ9+9MMk/LqW8PMlskk2llN+tta6KJkxf3ZHkjlrr0r8y/WEE6nHzg0m+XWvdmSSllE8k+QdJfnekVcUK9YqUUi5O8o4k/7jWum/U9dBXVyQ5u5RyZillOr2LHj454pros1JKSW/e8oZa66+Puh76r9b6rlrrqbXWM9L7Hv+VMD2eaq33JLm9lHLO4qGXJrl+hCXRf7cleX4pZd3i398vzSq58NQK9cr8VpKZJH/R+1zzpVrrW0dbEv1Qa22XUt6W5DPpXUX8gVrrdSMui/57YZIfT3JNKeVri8f+Za318hHWBDx5b0/ye4sLIbckefOI66GPaq1/X0r5wyRfSW/U9qtZJXdMdKdEAABYASMfAACwAgI1AACsgEANAAArIFADAMAKCNQAALACAjUAAKyAQA0AACsgUAM0RCnl+0op/3PZ6+eUUv5qlDUBjAM3dgFoiFLKmiR3JdlWa+2UUv46yS/WWr8y4tIAjmpuPQ7QELXWbinluiTPKKWcneQ2YRpg5QRqgGb5UpIXJvnpJBePuBaAsSBQAzTLl5J8MMl7a613jrgWgLFghhqgQRZHPf4mydm11r2jrgdgHNjlA6BZfi7Ju4RpgP4RqAEaoJTylFLKN5KsrbX+zqjrARgnRj4AAGAFrFADAMAKCNQAALACAjUAAKyAQA0AACsgUAMAwAoI1AAAsAICNQAArIBADQAAK/D/AzH/WdAsJ+vMAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "\n",
    "# Draw horizontal lines\n",
    "for xmin, xmax, y in [(-2, 1/5, 0), (1/3, 1, 3/4)]:\n",
    "    plt.hlines(y, xmin, xmax, color='C0')\n",
    "\n",
    "    \n",
    "# Draw increasing segments\n",
    "yy = np.arange(1/5, 1/3, step=0.001)\n",
    "plt.plot(yy, 15/8 - 3/(8 * yy), color='C0')\n",
    "\n",
    "yy = np.arange(1, 8, step=0.01)\n",
    "plt.plot(yy, 1 - 1/(4*yy), color='C0')\n",
    "    \n",
    "plt.xlabel(r'$y$')\n",
    "plt.ylabel(r'$F_Y(y)$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The probability density function is $f_y(y) = F'_Y(y)$, so\n",
    "\n",
    "$$ f_Y(y) = \\begin{cases}\n",
    "0  &\\text{if } y \\leq \\frac{1}{5} \\\\\n",
    "\\frac{3}{8y^2} &\\text{if } \\frac{1}{5} < y \\leq \\frac{1}{3} \\\\\n",
    "0 &\\text{if } \\frac{1}{3} < y \\leq 1 \\\\\n",
    "\\frac{1}{4y^2} &\\text{if } y > 1\n",
    "\\end{cases}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "def f_Y(y):\n",
    "    return np.where(y <= 1/5, 0, np.where(y <= 1/3, 3/(8 * (y**2)), np.where(y <= 1, 0, 1/(4 * (y**2)))))\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "\n",
    "yy = np.arange(-1, 3, step=0.001)\n",
    "plt.plot(yy, f_Y(yy), color='C0')\n",
    "    \n",
    "plt.xlabel(r'$y$')\n",
    "plt.ylabel(r'$f_Y(y)$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.5**.  Let $X$ and $Y$ be discrete random variables.  Show that $X$ and $Y$ are independent if and only if $f_{X, Y}(x, y) = f_X(x) f_Y(y)$ for all $x$ and $y$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  If $x$ or $y$ take values that have probability mass 0, then trivially $f_{X, Y}(x, y) = 0$ and $f_X(x) f_Y(y) = 0$, so we only need to consider $x$ and $y$ with positive probability mass.\n",
    "\n",
    "If $X$ and $Y$ are independent, then $\\mathbb{P}(X \\in A, Y \\in B) = \\mathbb{P}(X \\in A)\\mathbb{P}(Y \\in B)$ for all events $A$, $B$.  In particular, this is true for $A = \\{x\\}$ and $B = \\{ y \\}$ for all $x, y$, proving the implication in one direction.\n",
    "\n",
    "In the other direction, we have\n",
    "\n",
    "$$ \\mathbb{P}(X \\in A, Y \\in B) = \\sum_{x \\in A} \\sum_{y \\in B} f_{X, Y}(x, y) $$\n",
    "\n",
    "and\n",
    "\n",
    "$$ \n",
    "\\mathbb{P}(X \\in A) = \\sum_{x \\in A} f_X(x)\n",
    "\\quad \\text{and} \\quad\n",
    "\\mathbb{P}(Y \\in B) = \\sum_{y \\in B} f_Y(y)\n",
    "$$\n",
    "\n",
    "so $f_{X, Y}(x, y) = f_X(x) f_Y(y)$ for all $x, y$ implies that for all $A$ and $B$,\n",
    "\n",
    "$$ \\mathbb{P}(X \\in A) \\mathbb{P}(Y \\in B) = \\left( \\sum_{x \\in A} f_X(x) \\right) \\left( \\sum_{y \\in B} f_Y(y) \\right)\n",
    "= \\sum_{x \\in A} \\sum_{y \\in B} f_X(x) f_Y(y) = \\sum_{x \\in A} \\sum_{y \\in B} f_{X, Y}(x, y) = \\mathbb{P}(X \\in A, Y \\in B) $$\n",
    "\n",
    "so $X, Y$ are independent, proving the implication in the other direction."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.6**.  Let $X$ have distribution $F$ and density function $f$ and let $A$ be a subset of the real line.  Let $I_A(x)$ be the indicator function for $A$:\n",
    "\n",
    "$$\n",
    "I_A(x) = \\begin{cases}\n",
    "1 &\\text{if } x \\in A \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "Let $Y = I_A(X)$.  Find an expression for the cumulative distribution function of $Y$.  (Hint: first find the mass function for $Y$.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  Note that $I_A$ can only return values 0 or 1, so $Y$ is a discrete variable with non-zero probability mass only at values 0 and 1.\n",
    "\n",
    "But $\\mathbb{P}(Y = 1) = \\mathbb{P}(x \\in A) = \\int_A f_X(x) dx $, so the PDF for $Y$ is:\n",
    "\n",
    "$$ f_Y(y) = \\begin{cases}\n",
    "\\int_A f_X(x) dx &\\text{if } y = 1 \\\\\n",
    "1 - \\int_A f_X(x) dx &\\text{if } y = 0 \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases} $$\n",
    "\n",
    "and so the CDF of $Y$ is:\n",
    "\n",
    "$$\n",
    "F_Y(y) = \\begin{cases}\n",
    "0 &\\text{if } y < 0 \\\\\n",
    "1 - \\int_A f_X(x) dx &\\text{if } 0 \\leq y < 1 \\\\\n",
    "1 &\\text{if } y \\geq 1\n",
    "\\end{cases}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.7**.  Let $X$ and $Y$ be independent and suppose that each has a $\\text{Uniform}(0, 1)$ distribution.  Let $Z = \\min \\{ X, Y \\}$.  Find the density $f_Z(z)$ for $Z$.  Hint: it might be easier to first find $\\mathbb{P}(Z > z)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "$$1 - F_Z(z) = \\mathbb{P}(Z > z) = \\mathbb{P}(X > z, Y > z) = (1 - F_X(z)) (1 - F_Y(z)) $$\n",
    "\n",
    "But $F_X$ and $F_Y$ are both the CDF of the $\\text{Uniform}(0, 1)$ distribution, so\n",
    "\n",
    "$$ F_X(x) = \\begin{cases}\n",
    "0 &\\text{if } x \\leq 0 \\\\\n",
    "x &\\text{if } 0 < x \\leq 1 \\\\\n",
    "1 &\\text{if } x > 1\n",
    "\\end{cases} $$\n",
    "\n",
    "and so\n",
    "\n",
    "$$ F_Z(z) = \\begin{cases}\n",
    "0 &\\text{if } z \\leq 0 \\\\\n",
    "2z - z^2 &\\text{if } 0 < z \\leq 1 \\\\\n",
    "1 &\\text{if } z > 1\n",
    "\\end{cases} $$\n",
    "\n",
    "and the PDF is $f_Z(z) = F'_Z(z)$:\n",
    "\n",
    "$$ f_Z(z) = \\begin{cases}\n",
    "0 &\\text{if } z \\leq 0 \\\\\n",
    "2 - 2z &\\text{if } 0 < z < 1 \\\\\n",
    "0 &\\text{if } z > 1\n",
    "\\end{cases} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "def f_Z(z):\n",
    "    return np.where(z <= 0, 0, np.where(z >= 1, 0, 2 - 2 * z))\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "\n",
    "zz = np.arange(-0.5, 1.5, step=0.01)\n",
    "plt.plot(zz, f_Z(zz), color='C0')\n",
    "    \n",
    "plt.xlabel(r'$z$')\n",
    "plt.ylabel(r'$f_Z(z)$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.8**.  Let $X$ have CDF $F$.  Find the CDF of $X^+ = \\max \\{0, X\\}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "$$ F_{X^+}(x) = \\mathbb{P}(X^+ \\leq x) = \\mathbb{P}(0 < x, X < x) = I(0 < x) F_X(x) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.9**.  Let $X \\sim \\text{Exp}(\\beta)$.  Find $F(x)$ and $F^{-1}(q)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  Let's start from the definition for the PDF of the exponential distribution:\n",
    "\n",
    "$$ f(x) = \\frac{1}{\\beta} e^{-x / \\beta}, \\quad x > 0 $$\n",
    "\n",
    "We have:\n",
    "\n",
    "$$ F(x) = \\int_{-\\infty}^x f(t) dt = \\int_{-\\infty}^x \\frac{1}{\\beta} e^{-t / \\beta} dt = \\frac{1}{\\beta} \\beta \\left( 1 - e^{-x / \\beta} \\right) = 1 - e^{-x / \\beta} $$\n",
    "\n",
    "and so\n",
    "\n",
    "$$ q = 1 - e^{-F^{-1}(q) / \\beta} \\Longrightarrow F^{-1}(q) = -\\beta \\log (1 - q) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.10**.  Let $X$ and $Y$ be independent.  Show that $g(X)$ is independent of $h(Y)$ where $g$ and $h$ are functions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.   Let $g^{-1}(A) = \\{ x : g(x) \\in A \\}$ and $h^{-1}(B) = \\{ y : h(y) \\in B \\}$.  We have:\n",
    "\n",
    "$$\\mathbb{P}(g(X) \\in A, h(Y) \\in B) = \\mathbb{P}(X \\in g^{-1}(A), Y \\in h^{-1}(B) = \\mathbb{P}(X \\in g^{-1}(A)) \\mathbb{P}(Y \\in h^{-1}(B)) = \\mathbb{P}(g(X) \\in A) \\mathbb{P}(h(Y) \\in B)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 34.14.11**.  Suppose we toss a coin once and let $p$ be the probability of heads.  Let $X$ denote the number of heads and $Y$ denote the number of tails.\n",
    "\n",
    "**(a)** Prove that $X$ and $Y$ are dependent.\n",
    "\n",
    "**(b)** Let $N \\sim \\text{Poisson}(\\lambda)$ and suppose that we toss a coin $N$ times.  Let $X$ and $Y$ be the number of heads and tails.  Show that $X$ and $Y$ are independent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "**(a)**  The joint probability mass function is:\n",
    "\n",
    "$$\n",
    "\\begin{array}{c|cc}\n",
    "f_{X, Y} & Y = 0 & Y = 1 \\\\\n",
    "\\hline\n",
    "X = 0 & 0 & 1 - p\\\\\n",
    "X = 1 & p & 0\n",
    "\\end{array}\n",
    "$$\n",
    "\n",
    "In particular, $f_{X, Y}(1, 0) = p$ and $f_X(1) f_Y(0) = p^2$, so the events are dependent unless $p \\in \\{0, 1\\}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(b)**  The joint probability mass function is:\n",
    "\n",
    "$$ \\mathbb{P}(X = x, Y = y) = \\mathbb{P}(N = x + y, X = x) = \\frac{\\lambda^{x+y}e^{-\\lambda}}{(x + y)!} \\binom{x + y}{x} p^x (1-p)^y  = e^{-\\lambda} \\left(\\frac{\\lambda^x}{x!} p^x \\right) \\left(\\frac{\\lambda^y}{y!} (1 - p)^y\\right) = g(x) h(y)$$\n",
    "\n",
    "where $g(x) = e^{-\\lambda} \\frac{\\lambda^x}{x!} p^x $, $h(y) = \\frac{\\lambda^y}{y!} (1 - p)^y $.  The result then follows from theorem 3.33."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.12**.  Prove theorem 3.33.\n",
    "\n",
    "Suppose that the range of $X$ and $Y$ is a (possibly infinite) rectangle.  If $f(x, y) = g(x) h(y)$ for some functions $g$ and $h$ (not necessarily probability density functions) then $X$ and $Y$ are independent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "Given that $F$ is the joint probability density function of $X$ and $Y$, then the marginal distributions for $X$ and $Y$ are\n",
    "\n",
    "$$ f_X(x) = \\int f(x, y) dy = \\int g(x) h(y) dy = g(x) \\int h(y) dy = H g(x) \\quad \\text{ for some constant } H $$\n",
    "\n",
    "and\n",
    "\n",
    "$$ f_Y(y) = \\int f(x, y) dx = \\int g(x) h(y) dx = h(y) \\int g(x) dx = G h(y) \\quad \\text{ for some constant } G $$\n",
    "\n",
    "Therefore,\n",
    "\n",
    "$$ f_X(x) f_Y(y) = HG g(x) h(y) = HG f(x, y) $$\n",
    "\n",
    "Integrating over $x$, and fixing $y = y_0$ with non-zero probability density,\n",
    "\n",
    "$$ \\int f_X(x) f_Y(y_0) dx = HG \\int f(x, y_0) dx \\Longrightarrow f_Y(y_0) = HG f_Y(y_0) \\Longrightarrow HG = 1 $$\n",
    "\n",
    "and so \n",
    "\n",
    "$$ f_X(x) f_Y(y) = f(x, y) $$\n",
    "\n",
    "therefore $X$ and $Y$ are independent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.13**.  Let $X \\sim N(0, 1)$ and let $Y = e^X$.\n",
    "\n",
    "**(a)**  Find the PDF for $Y$.  Plot it.\n",
    "\n",
    "**(b) (Computer Experiment)**  Generate a vector $x = (x_1, \\dots, x_{10,000})$  consisting of 10,000 random standard Normals.  Let $y = (y_1, \\dots, y_{10,000})$ where $y_i = e^{x_i}$.  Draw a histogram of $y$ and compare it to the PDF you found in part (a)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "**(a)** \n",
    "\n",
    "Assuming $y > 0$,\n",
    "\n",
    "$$ F_Y(y) = \\mathbb{P}(Y \\leq y) = \\mathbb{P}(X \\leq \\log y) = \\Phi(\\log y) $$\n",
    "\n",
    "and so\n",
    "\n",
    "$$ f_Y(y) = \\frac{d}{dy} \\Phi(\\log y) = \\frac{d \\Phi(\\log y)}{d \\log y} \\frac{d \\log y}{dy} = \\frac{\\phi(\\log(y))}{y} $$\n",
    "\n",
    "And, of course, $Y$ can never take a negative value, so\n",
    "\n",
    "$$ f_Y(y) = \\begin{cases}\n",
    "\\frac{\\phi(\\log(y))}{y} &\\text{if } y > 0 \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "yy = np.arange(0.01, 5, step=0.01)\n",
    "\n",
    "def f_Y(y):\n",
    "    return norm.pdf(np.log(y)) / y\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "plt.plot(yy, f_Y(yy))\n",
    "plt.xlabel(r'$y$')\n",
    "plt.ylabel(r'$f_Y(y)$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(b)**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "\n",
    "N = 10000\n",
    "X = norm.rvs(size=N)\n",
    "Y = np.exp(X)\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "plt.hist(Y, bins=200, density=True, label='histogram', histtype='step')\n",
    "plt.plot(yy, f_Y(yy), label='true density')\n",
    "plt.legend(loc='upper right')\n",
    "plt.xlim(0, 5)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.14**.  Let $(X, Y)$ be uniformly distributed on the unit disc $\\{ (x, y) : x^2 + y^2 \\leq 1 \\}$.  Let $R = \\sqrt{X^2 + Y^2}$.  Find the CDF and the PDF of $R$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "Assuming $r > 0$,\n",
    "\n",
    "$$ F_R(r) = \\mathbb{P}(R \\leq r) = \\mathbb{P}(X^2 + Y^2 \\leq r^2) = \\int \\int_{x^2 + y^2 \\leq r^2} f(x, y) dx dy $$\n",
    "\n",
    "is proportional to the area of a disc of radius $r$.  Since $F_R(1) = 1$, we have that the CDF is\n",
    "\n",
    "$$ \n",
    "F_R(r) = \\begin{cases}\n",
    "0 &\\text{if } r \\leq 0 \\\\\n",
    "r^2 &\\text{if } 0 < r \\leq 1 \\\\\n",
    "1 &\\text{if } r > 1\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "and the PDF is $f_R(r) = F'_R(r)$:\n",
    "\n",
    "$$\n",
    "f_R(r) = \\begin{cases}\n",
    "0 &\\text{if } r \\leq 0 \\\\\n",
    "2 r &\\text{if } 0 < r \\leq 1 \\\\\n",
    "0 &\\text{if } r > 1\n",
    "\\end{cases}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.15 (A universal random number generator)**.  Let $X$ have a continuous, strictly increasing CDF.  Let $Y = F(X)$.  Find the density of $Y$.  This is called the probability integral transformation.  Now let $U \\sim \\text{Uniform}(0, 1)$ and let $X = F^{-1}(U)$.  Show that $X \\sim F$.  Now write a program that takes $\\text{Uniform}(0, 1)$ random variables and generates random variables from a $\\text{Exp}(\\beta)$ distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "For $0 \\leq y \\leq 1$, the CDF of $Y$ is\n",
    "\n",
    "$$ F_Y(y) = \\mathbb{P}(Y \\leq y) = \\mathbb{P}(F(X) \\leq y) = \\mathbb{P}(X \\leq F^{-1}(y)) = F(F^{-1}(y)) = y$$\n",
    "\n",
    "and so $Y \\sim \\text{Uniform}(0, 1)$.\n",
    "\n",
    "For $0 \\leq q \\leq 1$,\n",
    "\n",
    "$$ F_X(q) = \\mathbb{P}(X \\leq q) = \\mathbb{P}(F(X) \\leq F(q)) = \\mathbb{P}(U \\leq F(q)) = F(q) $$\n",
    "\n",
    "and so $X \\sim F$.\n",
    "\n",
    "To create a generator for $\\text{Exp}(\\beta)$, let $F$ be the CDF for this distribution,\n",
    "\n",
    "$$ F(x) = 1 - e^{-x/\\beta} \\quad F^{-1}(q) = - \\beta \\log (1 - q) $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "def inv_F(beta):\n",
    "    def inv_f_impl(q):\n",
    "        return -beta * np.log(1 - q)\n",
    "    return inv_f_impl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(0)\n",
    "\n",
    "N = 100000\n",
    "U = np.random.uniform(low=0, high=1, size=N)\n",
    "\n",
    "X = {}\n",
    "for beta in [0.1, 1.0, 10.0]:\n",
    "    X[beta] = inv_F(beta)(U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x576 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "from scipy.stats import expon\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "\n",
    "for i, beta in enumerate([0.1, 1.0, 10]):\n",
    "    ax = plt.subplot(3, 1, (i + 1))\n",
    "    ax.hist(X[beta], density=True, bins=100, histtype='step', label='simulation histogram')\n",
    "    xx = np.arange(beta * 0.01, 5 * beta, step=beta * 0.01)\n",
    "    ax.plot(xx, expon.pdf(xx, scale=beta), label='true PDF')\n",
    "    ax.set_xlim(-beta * 0.1, 5 * beta)\n",
    "    ax.legend(loc='upper right')\n",
    "    ax.set_title('beta = ' + str(beta))\n",
    "    \n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.16**.  Let $X \\sim \\text{Poisson}(\\lambda)$ and $Y \\sim \\text{Poisson}(\\mu)$ and assume that $X$ and $Y$ are independent.  Show that the distribution of $X$ given that $X + Y = n$ is $\\text{Binomial}(n, \\pi)$ where $\\pi = \\lambda / (\\lambda + \\mu)$.\n",
    "\n",
    "Hint 1: You may use the following fact:  If $X \\sim \\text{Poisson}(\\lambda)$ and $Y \\sim \\text{Poisson}(\\mu)$, and $X$ and $Y$ are independent, then $X + Y \\sim \\text{Poisson}(\\mu + \\lambda)$.\n",
    "\n",
    "Hint 2:  Note that $\\{X = x, X + Y = n\\} = \\{X = x, Y = n - x \\}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "We have:\n",
    "\n",
    "$$ \n",
    "\\begin{align}\n",
    "\\mathbb{P}(X = x | X + Y = n) &= \\frac{\\mathbb{P}(X = x, X + Y = n)}{\\mathbb{P}(X + Y = n)} \\\\\n",
    "&= \\frac{\\mathbb{P}(X = x, Y = n - x)}{\\mathbb{P}(X + Y = n)} \\\\\n",
    "&= \\frac{\\mathbb{P}(X = x) \\mathbb{P}(Y = n - x)}{\\mathbb{P}(X + Y = n)} \\\\\n",
    "&= \\frac{\\frac{\\lambda^x e^{-\\lambda}}{x!} \\frac{\\mu^{n - x} e^{-\\mu}}{(n - x)!} }{\\frac{(\\lambda + \\mu)^n e^{-(\\lambda + \\mu)}}{n!}} \\\\\n",
    "&= \\frac{n!}{x! (n - x)!} \\frac{\\lambda^x \\mu^{n - x}}{(\\lambda + \\mu)^n} \\\\\n",
    "&= \\binom{n}{x} \\left( \\frac{\\lambda}{\\lambda + \\mu} \\right)^x \\left( \\frac{\\mu}{\\lambda + \\mu}\\right)^{n - x} \\\\\n",
    "&= \\binom{n}{x} \\pi^x (1 - \\pi)^{n - x}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "and so the result follows."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.17**.  Let\n",
    "\n",
    "$$\n",
    "f_{X, Y}(x, y) = \\begin{cases}\n",
    "c(x + y^2) &\\text{if } 0 \\leq x \\leq 1 \\text{ and }  0 \\leq y \\leq 1 \\\\\n",
    "0 & \\text{otherwise}\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "Find $\\mathbb{P}\\left( X < \\frac{1}{2} | Y = \\frac{1}{2} \\right)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "The conditional density $f_{X | Y}(x | y)$ is:\n",
    "\n",
    "$$ f_{X|Y}(x | y) = \\frac{f_{X, Y}(x, y)}{f_Y(y)} = \\frac{f_{X, Y}(x, y)}{\\int f_{X, Y}(t, y) dt}\n",
    "= \\frac{ c(x + y^2) }{\\int_0^1 c(t + y^2) dt} =  \\frac{ x + y^2 }{\\int_0^1 t + y^2 dt} = \\frac{x + y^2}{\\frac{1}{2} + y^2} $$\n",
    "\n",
    "In particular, when $y = 1/2$,\n",
    "\n",
    "$$f_{X|Y}(x | 1/2) = \\frac{4x + 1}{3}$$\n",
    "\n",
    "and so\n",
    "\n",
    "$$ \\mathbb{P}\\left( X < x \\;\\Bigg|\\; Y = \\frac{1}{2} \\right) = \\int_0^x \\frac{4t + 1}{3} dt = \\frac{2x^2 + x}{3} $$\n",
    "\n",
    "so \n",
    "\n",
    "$$ \\mathbb{P}\\left( X < \\frac{1}{2} \\;\\Bigg|\\; Y = \\frac{1}{2} \\right) = \\frac{1}{3} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercie 3.14.18**.  Let $X \\sim N(3, 16)$.  Solve the following using the Normal table and using a computer package.\n",
    "\n",
    "**(a)** Find $\\mathbb{P}(X < 7)$.\n",
    "\n",
    "**(b)** Find $\\mathbb{P}(X > -2)$.\n",
    "\n",
    "**(c)** Find $x$ such that $\\mathbb{P}(X > x) = .05$.\n",
    "\n",
    "**(d)** Find $\\mathbb{P}(0 \\leq X < 4)$.\n",
    "\n",
    "**(e)** Find $x$ such that $\\mathbb{P}(|X| > |x|) = .05$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "Rather than using tables, we will express these in terms of a standard Normal, and use \"computer packages\" to compute the expression based both on the original Normal and the standard Normal."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(a)** $\\mathbb{P}(X < 7) = \\mathbb{P}\\left(\\frac{X - 3}{4} < \\frac{7 - 3}{4} \\right) = \\mathbb{P}\\left(Z < 1 \\right) = \\Phi\\left( 1 \\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.8413\n",
      "0.8413\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "\n",
    "print('%.4f' % norm.cdf(1))\n",
    "print('%.4f' % norm.cdf(7, loc=3, scale=4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(b)** $\\mathbb{P}(X > -2) = \\mathbb{P}\\left(\\frac{X - 3}{4} > \\frac{-2 - 3}{4} \\right) = \\mathbb{P}\\left(Z < -\\frac{5}{4} \\right) = 1 - \\Phi\\left( -\\frac{5}{4} \\right) $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.8944\n",
      "0.8944\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "\n",
    "print('%.4f' % (1 - norm.cdf(-5/4)))\n",
    "print('%.4f' % (1 - norm.cdf(-2, loc=3, scale=4)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(c)**  $\\mathbb{P}(X > x) = .05 \n",
    "\\Longleftrightarrow 1 - F_X(x) = .05 \n",
    "\\Longleftrightarrow F_X(x) = .95 \n",
    "\\Longleftrightarrow x = F_X^{-1}(.95)$\n",
    "\n",
    "or: $\\mathbb{P}(X > x) = .05 \n",
    "\\Longleftrightarrow \\mathbb{P}\\left(\\frac{X - 3}{4} > \\frac{x - 3}{4}\\right) = .05 \n",
    "\\Longleftrightarrow 1 - \\Phi\\left(\\frac{x - 3}{4}\\right) = .05 \n",
    "\\Longleftrightarrow \\frac{x - 3}{4} = \\Phi^{-1}(.95)\n",
    "\\Longleftrightarrow x = 4 \\Phi^{-1}(.95) + 3\n",
    "$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9.5794\n",
      "9.5794\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "\n",
    "print('%.4f' % norm.ppf(.95, loc=3, scale=4))\n",
    "print('%.4f' % (4 * norm.ppf(.95) + 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(d)** $\\mathbb{P}(0 \\leq X < 4) = F_X(4) - F_X(0)$\n",
    "\n",
    "or $\\mathbb{P}(0 \\leq X < 4) = \\mathbb{P}\\left(\\frac{0 - 3}{4} < Z < \\frac{4 - 3}{4} \\right) = \\Phi\\left( \\frac{1}{4} \\right) - \\Phi\\left( \\frac{-3}{4} \\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.3721\n",
      "0.3721\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "\n",
    "print('%.4f' % (norm.cdf(4, loc=3, scale=4) - norm.cdf(0, loc=3, scale=4)))\n",
    "print('%.4f' % (norm.cdf(1/4) - norm.cdf(-3/4)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(e)** $\\mathbb{P}(|X| > |x|) = .05$\n",
    "\n",
    "For a constant $c \\geq 0$, we have\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathbb{P}(|X| > c) &= 1 - \\mathbb{P}(|X| < c) = 1 - \\mathbb{P}(-c < X < -c) \\\\\n",
    "&= 1 - \\mathbb{P}\\left( -\\frac{c - 3}{4} < Z < \\frac{c - 3}{4} \\right)\\\\\n",
    "&= 1 - \\Phi\\left( \\frac{c - 3}{4} \\right) + \\Phi\\left( -\\frac{c - 3}{4} \\right) \\\\\n",
    "&= 2 - 2 \\Phi\\left( \\frac{c - 3}{4} \\right) = 2 - 2 F_X(c)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "So we can solve the original equation:\n",
    "\n",
    "$$ 2 - 2 F_X(c) = .05 \\Longleftrightarrow c = F_X^{-1}(0.975) $$\n",
    "\n",
    "or\n",
    "\n",
    "$$ 2 - 2 \\Phi\\left( \\frac{c - 3}{4} \\right) = .05 \\Longleftrightarrow c = 4 \\Phi^{-1}(0.975) + 3 $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10.8399\n",
      "10.8399\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "\n",
    "print('%.4f' % (norm.ppf(0.975, loc=3, scale=4)))\n",
    "print('%.4f' % (4 * norm.ppf(0.975) + 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.19**.  Prove formula (3.11).\n",
    "\n",
    "Given $Y = r(X)$, when $r$ is strictly monotone increasing or strictly monotone decreasing then $r$ has an inverse $s = r^{-1}$ and in this case one can show that\n",
    "\n",
    "$$ f_Y(y) = f_X(s(y)) \\;\\Bigg| \\frac{ds(y)}{dy} \\Bigg|$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "Assume $r$ is strictly monotone increasing. Then $ \\frac{d s(y)}{dy} > 0$, and the CDF of $Y$ is\n",
    "\n",
    "$$ F_Y(y) = \\mathbb{P}(Y \\leq y) = \\mathbb{P}(r(X) \\leq y) = \\mathbb{P}(X \\leq s(y)) = F_X(s(y)) $$\n",
    "\n",
    "Derivating over $y$,\n",
    "\n",
    "$$ \n",
    "\\begin{align}\n",
    "\\frac{d}{dy} F_y(y) &= \\frac{d F_X(s(y))}{d s(y)} \\frac{d s(y)}{dy} \\\\\n",
    "f_Y(y) &= f_X(s(y)) \\frac{d s(y)}{dy}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "For the case when $r$ is strictly monotone decreasing,  $ \\frac{d s(y)}{dy} < 0$, and the CDF of $Y$ is\n",
    "\n",
    "$$ F_Y(y) = \\mathbb{P}(Y \\leq y) = \\mathbb{P}(r(X) \\leq y) = \\mathbb{P}(X \\geq s(y)) = 1 - F_X(s(y)) $$\n",
    "\n",
    "Derivating over $y$,\n",
    "\n",
    "$$ \n",
    "\\begin{align}\n",
    "\\frac{d}{dy} F_y(y) &= \\frac{d -F_X(s(y))}{d s(y)} \\frac{d s(y)}{dy} \\\\\n",
    "f_Y(y) &= -f_X(s(y)) \\frac{d s(y)}{dy}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "and the result follows."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.20**.  Let $X, Y \\sim \\text{Uniform}(0, 1)$ be independent.  Find the PDF for $X - Y$ and $X / Y$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "The joint density of $X, Y$ is\n",
    "\n",
    "$$ \n",
    "f(x, y) = \\begin{cases}\n",
    "1 &\\text{if } 0 \\leq x \\leq 1, 0 \\leq y \\leq 1 \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "Let $A = X - Y$.  The CDF of $A$ is calculated over the area $A_a = \\{ (x, y) : x - y \\leq a \\}$:\n",
    "\n",
    "$$ F_A(a) = \\mathbb{P}(X + Y \\leq a) = \\int_{A_a} f(x, y)\\; dx dy $$\n",
    "\n",
    "- If $0 \\leq a \\leq 1$, the area is consists of all points in the unit square, othe than the triangle at points $(a, 0), (1, 0), (1, 1 - a)$.  Then,\n",
    "\n",
    "  $$ F_A(a) = 1 - \\frac{(1 - a)^2}{2} $$\n",
    "\n",
    "- If $-1 \\leq a \\leq 0$, the area consists of only the points in the triangle $(0, 1), (1 + a, 1), (0, -a)$.  Then,\n",
    "\n",
    "  $$ F_A(a) = \\frac{(1 + a)^2}{2} $$\n",
    "  \n",
    "Therefore, the PDF is $f_A(a) = F'_A(a)$, or\n",
    "\n",
    "$$ \n",
    "f_A(a) = \\begin{cases}\n",
    "1 + a &\\text{if } -1 < a \\leq 0 \\\\\n",
    "1 - a &\\text{if } 0 < a \\leq 1 \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases} \n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.figure(figsize=(8, 4))\n",
    "\n",
    "ax = plt.subplot(1, 2, 1)\n",
    "a = 1/3\n",
    "x = [0, a, 1, 1, 0]\n",
    "y = [0, 0, 1 - a, 1, 1]\n",
    "ax.fill(x, y)\n",
    "ax.set_xlim(-0.5, 1.5)\n",
    "ax.set_ylim(-0.5, 1.5)\n",
    "ax.set_title('a = 1/3')\n",
    "\n",
    "ax = plt.subplot(1, 2, 2)\n",
    "a = -1/3\n",
    "x = [0, 1+a, 0]\n",
    "y = [1, 1, -a]\n",
    "ax.fill(x, y)\n",
    "ax.set_xlim(-0.5, 1.5)\n",
    "ax.set_ylim(-0.5, 1.5)\n",
    "ax.set_title('a = -1/3')\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $B = X/Y$.  The CDF of $B$ is calculated over the area $B_b = \\left\\{ (x, y) : \\frac{x}{y} \\leq b \\right\\}$.\n",
    "\n",
    "$$ F_B(a) = \\mathbb{P}\\left(\\frac{X}{Y} \\leq b\\right) = \\int_{B_b} f(x, y)\\; dx dy $$\n",
    "\n",
    "- If $0 < b \\leq 1$, the relevant area is a triangle with points $(0, 0), (1, 0), (1, b)$.  Then,\n",
    "\n",
    "  $$ F_B(b) = \\frac{b}{2} $$\n",
    "  \n",
    "- If $b > 1$, the relevant area is the unit square minus the triangle with points $(0, 0), (0, 1), (1/b, 1)$.  Then,\n",
    "\n",
    "  $$ F_B(b) = 1 - \\frac{1}{2b} $$\n",
    "  \n",
    "Therefore, the PDF is $f_B(b) = F'_B(b)$, or\n",
    "\n",
    "$$ \n",
    "f_A(a) = \\begin{cases}\n",
    "\\frac{1}{2} &\\text{if } 0 < b \\leq 1 \\\\\n",
    "\\frac{1}{2b^2} &\\text{if } b > 1 \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases} \n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.figure(figsize=(8, 4))\n",
    "\n",
    "ax = plt.subplot(1, 2, 1)\n",
    "b = 1/3\n",
    "x = [0, b, 0]\n",
    "y = [0, 1, 1]\n",
    "ax.fill(x, y)\n",
    "ax.set_xlim(-0.5, 1.5)\n",
    "ax.set_ylim(-0.5, 1.5)\n",
    "ax.set_title('b = 1/3')\n",
    "\n",
    "ax = plt.subplot(1, 2, 2)\n",
    "b = 3\n",
    "x = [0, 1, 1, 0]\n",
    "y = [0, 1/b, 1, 1]\n",
    "ax.fill(x, y)\n",
    "ax.set_xlim(-0.5, 1.5)\n",
    "ax.set_ylim(-0.5, 1.5)\n",
    "ax.set_title('b = 3')\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.21**.  Let $X_1, \\dots, X_n \\sim \\text{Exp}(\\beta)$ be IID.  Let $Y = \\max \\{ X_1, \\dots, X_n \\}$.  Find the PDF of $Y$.  Hint: $Y \\leq y$ if and only if $X_i \\leq y$ for $i = 1, \\dots, n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  We have:\n",
    "\n",
    "$$ \\mathbb{P}(Y \\leq y) = \\mathbb{P}(\\forall i, X_i \\leq y) = \\prod_i \\mathbb{P}(X_i \\leq y) = \\prod_i \\left(1 - e^{-y/\\beta}\\right) = (1 - e^{-y/\\beta})^n $$\n",
    "\n",
    "and so the PDF is $f_Y(y) = F'_Y(y)$:\n",
    "\n",
    "$$ f_Y(y) = \\frac{d (1 - e^{-y/\\beta})^n}{dy} = \\frac{n}{\\beta} e^{-y/\\beta} (1 - e^{-y/\\beta})^{n-1}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.22**.  Let $X$ and $Y$ be random variables. Suppose that $\\mathbb{E}(Y | X) = X$.  Show that $\\text{Cov}(X, Y) = \\mathbb{V}(X)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "The covariance is:\n",
    "\n",
    "$$ \\text{Cov}(X, Y) = \\mathbb{E}(XY) - \\mathbb{E}(X) \\mathbb{E}(Y) $$\n",
    "\n",
    "and the variance is\n",
    "\n",
    "$$ \\mathbb{V}(X) = \\mathbb{E}(X^2) - \\mathbb{E}(X)^2 $$\n",
    "\n",
    "We aim to prove that both of those expressions have the same value when $\\mathbb{E}(Y | X) = X$.\n",
    "\n",
    "Taking the expectaion on $\\mathbb{E}(Y | X)$ we get:\n",
    "\n",
    "$$ \\mathbb{E}(\\mathbb{E}(Y | X)) = \\int \\mathbb{E}(Y = y | X) f_Y(y) \\; dy = \\int y f_Y(y) = \\mathbb{E}(Y) $$\n",
    "\n",
    "so taking the expectation on both sides of $\\mathbb{E}(Y | X) = X$ gives us $\\mathbb{E}(Y) = \\mathbb{E}(X)$.\n",
    "\n",
    "Now, we have\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathbb{E}(XY) &= \\int \\mathbb{E}(XY | X = x) f_X(x) dx \\\\\n",
    "&= \\int \\mathbb{E}(xY | X = x) f_X(x) dx \\\\\n",
    "&= \\int x \\mathbb{E}(Y | X = x) f_X(x) dx \\\\\n",
    "&= \\int x^2 f_X(x) dx \\\\\n",
    "&= \\mathbb{E}(X^2)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Using $\\mathbb{E}(X^2) = \\mathbb{E}(XY)$ and $\\mathbb{E}(Y) = \\mathbb{E}(X)$, we get the desired result."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.14.23**.  Let $X \\sim \\text{Uniform}(0, 1)$.  Let $0 < a < b < 1$.  Let\n",
    "\n",
    "$$\n",
    "Y = \\begin{cases}\n",
    "1 &\\text{if } 0 < x < b \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "and let\n",
    "\n",
    "$$\n",
    "Z = \\begin{cases}\n",
    "1 &\\text{if } a < x < 1 \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "**(a)**  Are $Y$ and $Z$ independent?  Why / Why not?\n",
    "\n",
    "**(b)**  Find $\\mathbb{E}(Y | Z)$.  Hint: what values $z$ can $Z$ take?  Now find $\\mathbb{E}(Y | Z = z)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "**(a)**  The joint probability mass function for $Y$ and $Z$ is:\n",
    "\n",
    "$$\n",
    "\\begin{array}{c|cc}\n",
    " & Z = 0 & Z = 1 \\\\\n",
    "\\hline\n",
    "Y = 0 & 0 & 1 - b \\\\\n",
    "Y = 1 & a & b - a\n",
    "\\end{array}\n",
    "$$\n",
    "\n",
    "These are not independent; in particular,\n",
    "\n",
    "$$ \\mathbb{P}(Y = 1, Z = 1) = b - a \\neq \\mathbb{P}(Y = 1) \\mathbb{P}(Z = 1) = b(1 - a) $$\n",
    "\n",
    "since the equality would only hold if $b = 0$ or $a = 0$, and the inequality precludes both of these."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(b)**  We have:\n",
    "\n",
    "$$ \\mathbb{E}(Y | Z = 0) = \\frac{\\sum_y y f(y, 0)}{f_Z(0)} = \\frac{f(1, 0)}{a} = 1 $$\n",
    "\n",
    "and\n",
    "\n",
    "$$ \\mathbb{E}(Y | Z = 1) = \\frac{\\sum_y y f(y, 1)}{f_Z(1)} = \\frac{f(1, 1)}{1 - a} = \\frac{b - a}{1 - a} $$"
   ]
  }
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