Commonly used discrete random distribution

Commonly used discrete random variable distribution and its correlation function

In the third party Library scipy in , It includes the distribution commonly used in Statistics . Mainly in the scipy.stats Module , So before using it , Except to install scipy Outside , Also import the distribution of responses .

One scipy.stats Discrete distribution commonly used in

• Bernoulli distribution (Bernoulli)
• The binomial distribution (Binomial)
• Poisson distribution (Poisson)
• Geometric distribution (Geometri)

Two Methods common to random variables

• Probability Mass Function(pmf) Probability mass function , Straight white point , Is to find the probability of taking a given sample point .

$p(x_k)=P(X=x_k)$

• Cumulative Distribution Function(cdf) Cumulative distribution function

  $F(x)=P(X\le x)={\sum }_{x_k\le x}p(x_k)$

• Survival Function(sf) Survival function

  $S(x)=1-F(x)=P(X>k)$

• Percent Point Function(ppf) Percentiles , ppf yes cdf The inverse function of

• Inverse survival Function(isf) Inverse function of survival function .

3、 ... and Python The use of random variable distribution commonly used in

3.1 Freeze a distribution

Each distribution has fixed parameters , If these parameters are transmitted every time when using this distribution , Obviously it's more troublesome . Therefore, you can first pass in the corresponding parameters , That is to freeze this distribution .

>>> from scipy import stats
>>> rv1 = stats.binom(100,0.05) #  Freeze one `n=100, p=0.05` The binomial distribution of 
>>> rv2 = stats.poisson(5)  #  Freeze a parameter λ=5 The Poisson distribution of 
>>> rv3 = stats. geom(0.5)  #  Freeze a parameter p=5 The geometric distribution of 

3.2 The use of correlation functions

$X\sim B(10,0.05)$ , seek $P(x=10),P(x\le 10)$

>>> from scipy import stats
>>> rv1 = stats.binom(100, 0.05) #  Frozen distribution 
>>> rv1.pmf(10) #  Find the probability of the given value of random variables 
0.016715884095931225
>>> rv1.cdf(10) #  Find the cumulative probability 
0.9885275899325152

$X\sim P(5)$, seek $P(x=10),P(x\le 10)$

>>> from scipy import stats
>>> rv2 = stats.poisson(5) #  Frozen distribution 
>>> rv2.pmf(10)
0.018132788707821854
>>> rv2.cdf(10)
0.9863047314016171

Four Comparison between binomial distribution and Poisson distribution

In binomial distribution , When n It's big , p In small cases , Poisson distribution can be approximately used to calculate .

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

def binomAndpoisson(n, p):
    #  Frozen distribution 
    lambda1 = n*p
    rv1 = stats.binom(n, p)
    rv2 = stats.poisson(lambda1)

    x = np.arange(n)
    width = .35

    pbs=rv1.pmf(x)
    pps = rv2.pmf(x)

    fig, ax = plt.subplots()
    rects1 = ax.bar(x-width/2, pbs, width, label="binom")
    rects2 = ax.bar(x+width/2, pps, width, label="poisson")

    ax.set_ylabel('Probability')
    ax.set_title("Compare binom(n={},p={}) with poisson({} )".format(n,p, n*p))
    ax.set_xticks(x)
    ax.legend()

    # ax.bar_label(rects1, padding=3)
    # ax.bar_label(rects2, padding=3)

    fig.tight_layout()
    plt.show()

 binomAndpoisson(5, 0.2)
# binomAndpoisson(50, 0.2)
# binomAndpoisson(100, 0.02)

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It can be seen from the test on the summary , When the binomial distribution is n It's big , p Very hour , It can be used $\lambda =np$ To approximate the Poisson distribution of .

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It can be seen from the test on the summary , When the binomial distribution is n It's big , p Very hour , It can be used $\lambda =np$ To approximate the Poisson distribution of .

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