MCS: multivariate random variable polynomial distribution

Multinomial polynomial

Suppose an experiment has $k$ Two independent results :$R_1,R_2,...R_k$ The corresponding probabilities of occurrence are ：$p_1,p_2,...p_k$. And ${\sum }_{i=1}^k{p}_i=1.0$. Independent repetition $n$ Experiments , The number of times each experimental result occurs can be expressed as a random variable $x_1,x_2,..x_k$ To express ,${\sum }_{i=1}^k{x}_i=n$.

$x_i$ Probability distribution of ：

$$P(x_1,...,x_k)=\frac{n!}{[x_1!...x_k!]p_1^{x_1}...p_k^{x_k}}$$

$x_i$ On the edge of （ Expectation and variance ）：

$$E(x_i)=n{p}_i$$
$$V(x_i)=n{p}_i(1-p_i)$$

Generate polynomial random variables

It is known that ： Multiple display variables :$(x_1,x_2,...,x_k)$, Probability of occurrence :$(p_1,P_2,...p_k)$

1. $i=1\to k$
   • $p_i^{\prime }=p_i/{\sum }_{j=i}^k{p}_j$
   • $n_i^{\prime }=n-{\sum }_{j=1}^{i-1}{x}_j$
   • Generate a random binomial variable ：$x_i\sim Binomial(n_i^{\prime },p_i^{\prime })$
2. $(x1,x_2,...x_k)$

example ： Suppose there are only three possible outcomes in an experiment , The probabilities are :0.5, 0.3, 0.2. Suppose you repeat five independent experiments , The number of times each situation occurs in this case , namely ： Random polynomial variables ：$x_1,x_2,x_3$ How much could it be ？

1. $i=1,n_1^{\prime }=5,p_1^{\prime }=0.5,Binomial(5,0.5)=2,x_1=2$
2. $i=2,n_2^{\prime }=3,p_2^{\prime }=0.3/(0.3+0.2)=0.6,Binomial(3,0.6)=2,x_2=2$
3. $i=3,n_3^{\prime }=1,p_3^{\prime }=0.2/0.2=1.0,Binomial(1,1.0)=1,x_3=1$
4. $(x_1=2,x_2=2,x_3=1)$

Simulation generates polynomial variables

import numpy as np
import matplotlib.pyplot as plt

def generateMultinomial(n = 100, k=3, probas=[0.5, 0.3, 0.2]):
    x = [0, 0, 0]
    for i in range(k):
        p_ = probas[i]/np.sum(probas[i:])
        n_ = n - np.sum(x[:i+1])
        b = np.random.binomial(n_, p_)
        x[i] = b
    return x