Issues with Exercise 5.6.4 b) (Chapter 5: Inequalities)

[vashcast]

As pointed out by the statement of the question $\mathbb{P}(p \in C_n) \geq 1- \alpha = 0.95$, for $\alpha=0.05$. According to this, the Coverage should always be above 0.95, which is not what we see in the simulation provided here.

Sincerely, I do not understand the simulation provided since I am not familiar with "tqdm" and maybe other things. If it helps, what I did was to generate 100 Bernoulli(p) RV of length N=10.000, calculate the 100 cumulative RVs (of length N=10000) and average the coverage over the 100 repetitions. My caveats: I am not sure I really need to average over a given number of random vectors (which I arbitrarily decided to be 100)

Here mycode:

```
import math
import random as rd
import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import bernoulli
np.random.seed(1)

N = 10000
p = 0.4
alpha=0.05
nn=np.arange(1,N+1)
XX_Cum=[np.cumsum(bernoulli.rvs(p,size=N))/nn for i in range(100)]

def epsilon(N,alpha):
    return np.sqrt(1/(2*N)*np.log(2/alpha))

Counter=np.array([[np.abs(XX_Cum[j][i-1]-p)<=epsilon(i,alpha) for i in range(1,N+1)] for j in range(100)])

Coverage=[np.count_nonzero(Counter.transpose()[i])/100 for i in range(len(Counter.transpose()))]

plt.plot(nn, Coverage,label='Coverage')
plt.plot(nn, [1-alpha for i in range(len(Counter.transpose()))],label='Hoeffding s lower bound')
# naming the x axis
plt.xlabel('N')
# naming the y axis
plt.ylabel('$P(p \in C_n)$')
# giving a title to my graph
plt.legend()
  
# function to show the plot
plt.show()

```