【Python】2 examples of Chebyshev inequality.

2017年08月04日

Author: Guofei

文章归类: 趣文，文章编号:

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About the theory of Chebyshev inequality, see law of large numbers.

2 examples

This blog gives 2 examples of Chebyshev inequality: standard norm distribution, student’s t-distribution.

chebyshev

Here are two forms of Chebyshev inequality:

chebyshev1

Pr[∣X−u∣≥s]≤σ2s2Pr[∣X−u∣≥s]≤σ2s2
If the p.d.f is an even function ( meaning u=0 ),
Pr[X−u≥s]≤σ22s2Pr[X−u≥s]≤σ22s2

chebyshev2

(also known as Cantelli’s inequality)
Pr[X−u≥s]≤σ2s2+σ2Pr[X−u≥s]≤σ2s2+σ2

let u=0,σ2=1u=0,σ2=1

def chebyshev1(u,sigma,s):
    return sigma**2/s**2

def chebyshev2(u,sigma,s):
    return sigma**2/(sigma**2+s**2)

import numpy as np
u=0;sigma=1;

chebylist1=[]
chebylist2=[]
xlist=np.arange(0.5,5,0.01)
for s in xlist:
    #min(chebyshev1(u,sigma,s)/2,chebyshev2(u,sigma,s))
    chebylist1.append(chebyshev1(u,sigma,s)/2)
    chebylist2.append(chebyshev2(u,sigma,s))


from scipy.stats import norm
normlist=norm.sf(xlist)

from scipy.stats import t
tlist=t.sf(xlist,df=2)

import matplotlib.pyplot as plt
plt.plot(xlist,chebylist1)
plt.plot(xlist,chebylist2)
plt.plot(xlist,normlist)
plt.plot(xlist,tlist)
plt.legend(['chebyshev inequality 1','chebyshev inequality 2','norm distribution equality','t-distribution equality'])
plt.show()

output:

The graph tells us the accuracy of the Chebysheve inequality.

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【Python】2 examples of Chebyshev inequality.