This document defines functions to calculate various probability bounds for binomial distributions. It generates random binomial samples, calculates upper bounds on tail probabilities using Chernoff, Hoeffding, Markov, and Chebyshev bounds, and plots the results. It finds that numerically, P(|X-n/2|>=sqrt(n)/2) approaches 1/2 and P(X>=(n+sqrt(n))/2) approaches 1/4 for large n. It also compares the Chebyshev and Hoeffding bounds for various values of x.

1. import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import norm,binom,poisson,uniform,expon
from math import sqrt,exp
from scipy.integrate import quad
def uni2(n,N,s):
x1,x2 = 0,0
for i in range(N):
if s[i]>= (n+sqrt(n))/2: #P(X >= n+sqrt(n))/2)
x1 += 1
x2 = x1/N
return x2
def uni(n,N,s):
x3,x4 = 0,0,
for i in range(N):
if np.abs(s[i]-u)>= sqrt(n)/2:
x3 += 1
x4=x3/N
return x4
def uni3(n,N,s,x):
x5,x6=0,0
for i in range(N):
if np.abs(s[i]-u)>= x*sqrt(n):
x5+=1
x6=x5/N
return x6
def chebyshev(u,sigma,n):
return sigma/((sqrt(n))/2)**2
chebylist=[]
def markov(u,n):
return u/((n+sqrt(n))/2)
markovlist=[]
def chernoff(v,n,t):
return v/exp(t*((n+sqrt(n))/2))
chernofflist = []
def hoeffding(x):
return 2*exp(-x**2)
def chebyshev2(p,x):
return p*(1-p)/x**2

2. ###### X suit B(n,1/2)
N= 100
p=0.5
t=0.5
x= 1
xlist = np.arange(0.5,100,0.1)
binom1=[]
binom2=[]
binom3=[]
chernofflist = []
chebylist=[]
chebylist2=[]
markovlist=[]
hoeffdinglist=[]
for n in xlist:
u=n*p
s = np.random.binomial(n,p,N)
sigma = n*p*(1-p)
v = (exp(t)*p + (1-p))**n
binom1.append(uni(n,N,s))
binom2.append(uni2(n,N,s))
binom3.append(uni3(n,N,s,x))
chebylist.append(chebyshev(u,sigma,n))
markovlist.append(markov(u,n))
chernofflist.append(chernoff(v,n,t))
hoeffdinglist.append(hoeffding(x))
chebylist2.append(chebyshev2(p,x))
plt.plot(xlist,binom1,label='P(|X- 'r'$frac {n}{2}| geqslantfrac {sqrt{n}} {2})$')
plt.plot(xlist,chebylist,label = 'Chebyshev')
plt.xlabel('n')
plt.ylabel('Probabilité')
plt.legend()
plt.show()
###############Chebyshev = 1 inutile et on remarque numériquement que : P(|X-n/2|⩾√n/2)⩽1/2
########*Chernoff et markov loin d'être optimaux et on remarque numériquement que :
P(X⩾(n+√n)/2)⩽1/4
plt.plot(xlist,binom2,label='P(X'r'$ geqslant frac {n+sqrt{n}} {2})$')
plt.plot(xlist,chernofflist, label = 'Chernoff')
plt.plot(xlist,markovlist,label = 'Markov')

3. plt.xlabel('n')
plt.ylabel('Probabilité')
plt.ylim(ymax=1)
plt.legend()
plt.show()
##############
### P(|X- n/2|⩾x√n)⩽p(1−p)/x²: Chebyshev
###P(|X- n/2|⩾x√n)⩽2exp(−2x²): Hoeffding
plt.plot(xlist,binom3,label='P(|X- 'r'$frac {n}{2}| geqslant xsqrt{n})$' )
plt.plot(xlist,hoeffdinglist,label = 'Hoeffding')
plt.plot(xlist,chebylist2,label = 'Chebyshev')
plt.xlabel('n')
plt.ylabel('P(|X- 'r'$frac {n}{2}| geqslant xsqrt{n})$')
plt.legend()
plt.show()
####### Comparaison Chebyshev et Hoeffding en fonction de x
blist = np.arange(0.5,10,0.1)
chebylist3=[]
hoeffdinglist2=[]
for x in blist :
hoeffdinglist2.append(hoeffding(x))
chebylist3.append(chebyshev2(p,x))
plt.plot(blist,hoeffdinglist2,label = 'Hoeffding')
plt.plot(blist,chebylist3,label = 'Chebyshev')
plt.xlabel('x')
plt.legend()
plt.show()