In [1]:

    

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom


    

In [2]:

    

'''Computes the cumulative density function of min{X,Y}'''
def cdf_min(X,Y):
    return 1 - (1-X)*(1-Y)

'''Computes the probability mass function from the cumulative density'''
def cdf_to_pmf(X):
    return np.concatenate(([X[0]], np.diff(X)))

'''Constructs the cdf for the shift distance t with paramter n and p'''
def shift_distance_cdf(t, n, p):
    S = binom.cdf(range(n), n-1, p)
    S = np.concatenate(([0]*t*2, S))
    return S

'''Computes the cdf of the minimum of several cdfs'''
def cdf_min_a(M):
    m = M.shape[0]
    X = M[0,:]
    for i in range(1,m):
        X = cdf_min(X, M[i,:])
    return X


    

In [3]:

    

p = 3/4
ns = [16]
for n in ns:
    Ks = range(n+1)
    X = binom.cdf(range(n+1), n, p)
    Z = cdf_min(X,X)
    X_pmf = cdf_to_pmf(X)
    Z_pmf = cdf_to_pmf(Z)
    print("E[X] = {0}".format(np.sum(X_pmf*Ks)))
    print("E[Z] = {0}".format(np.sum(Z_pmf*Ks)))
    plt.plot(Ks, X_pmf, label='X')
    plt.plot(Ks, Z_pmf, label='min{X,X}')
    plt.legend()


    

    




E[X] = 12.0
E[Z] = 11.035657741699676






    

In [4]:

    

ns = range(16,4099,32)
As = []
for n in ns:    
    Ks = range(n+1)
    SDs = binom.cdf(Ks, n, p)
    ts = [1,2]
    for t in ts:
        St = shift_distance_cdf(t, n, p)
        St = St[:n+1]
        Z = cdf_min(St, St)
        SDs = np.vstack((SDs, Z))
        St_pmf = cdf_to_pmf(St)
        Z_pmf = cdf_to_pmf(Z)        
    SD_cdf = cdf_min_a(SDs)
    SD_pmf = cdf_to_pmf(SD_cdf)
    #print("n = {0}\n  E[SD] = {1:.2f}\n  E[X] =  {2}".format(n, np.sum(Ks*SD_pmf), n*p))
    a_n = np.sum(Ks*SD_pmf)/n
    As.append(a_n)
    #print("  alpha_n = {0}".format(a_n))
plt.plot(ns,As)


    

    
Out[4]:





[<matplotlib.lines.Line2D at 0x7fe7f81739e8>]






    

In [5]:

    

min(As)


    

    
Out[5]:





0.699946510845151

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