MCS: multivariate random variable - discrete random variable

Multivariate Discrete Arbitrary

Yes $k$ A discrete random variable ：$(x_1,...,x_k)$, The joint probability distribution is ：$P(x_1,...,x_k)$,$k$ The probability sum of all possible values of variables is :1.
$$\sum _{x_1\sim x_k}P(x_1...x_k)=1.0$$

Marginal probability ,$k$ One of the variables , Write it down as ：$x_j$

$$P(x_j...)=\sum _{al{l}_{x}bu{t}_{x_j}}P(x_1,x_2,...,x_k)$$

Marginal expectation ：

$$E(x_j...)=\sum _{x_j}{x}_{j}P(x_j...)$$

marginal Variance ：

$$V(x_j...)=E(x_j^2...)-E(x_j...{)}^2$$

$$E(x_j^2...)=\sum _{x_j}{x}_j^{2}P(x_j...)$$

Generating random variables

Generate $k$ A random variable of discrete variables ：$(x_1,x_2,...,x_k)$

1. obtain $x_1$ Marginal probability and cumulative distribution ： $P(x_1...)、F(x_1...)$
   • Generate a random continuous uniform variable ：$u\sim U(0,1)$
   • Find a random variable larger than $u$ Of $F(x_1...)$ Corresponding $x_1$ The minimum value of , Write it down as ：$x_{10}$
2. obtain $x_2$ Marginal conditional probability and cumulative distribution ：$P(x_2|x_{10}...)、F(x_2|x_{10}...)$
   • Generate a random continuous uniform variable ：$u\sim U(0,1)$
   • Find a random variable larger than $u$ Of $F(x_2|x_{10}...)$ Corresponding $x_2$ The minimum value of , Write it down as ：$x_{20}$
3. obtain $x_3$ Marginal conditional probability and cumulative distribution ：$P(x_3|x_{10}{x}_{20}...)、F(x_3|x_{10}{x}_{20}...)$
   • Generate a random continuous uniform variable ：$u\sim U(0,1)$
   • Find a random variable larger than $u$ Of $F(x_3|x_{10}{x}_{2}0...)$ Corresponding $x_3$ The minimum value of , Write it down as ：$x_{30}$
4. Repeat until you get ：$x_{k0}$
5. $(x_{1}0,...x_{k0})$

example ： Suppose a multivariate discrete random variable :$(x_1,x_2,x_3)$,$x_1$ Possible values are ：$\{0,1,2\}$,$x_2$ Possible values are ：$\{0,1\}$,$x_3$ Possible values are ：$\{1,2,3\}$, The probability distribution is as follows ：
$$P(x_1,x_2,x_3)$$

$x_3$	1	2	3	1	2	3
$x_2$	0			1
$x_1$
0	0.12	0.10	0.08	0.08	0.06	0.05
1	0.08	0.06	0.04	0.05	0.04	0.03
2	0.06	0.04	0.02	0.04	0.03	0.02

Generate multivariate discrete random variables according to probability distribution ：

1. $x_1$ Marginal probability and cumulative distribution ：
   • $P(x_1=0...)=0.12+0.10+0.08+0.08+0.06+0.05=0.49、F(0...)=0.49$
   • $P(x_1=1...)=0.08+0.06+0.04+0.05+0.04+0.03=0.30、F(1...)=0.79$
   • $P(x_1=2...)=0.06+0.04+0.02+0.04+0.03+0.02=0.21、F(2...)=1.00$
   • $u\sim U(0,1),u=0.37,u<F(0)$ Make ：$x_{10}=0$
2. $x_2$ Of ( Conditions ) Marginal probability and cumulative distribution ：
   • $P(x_2=0|x_{10}=0...)=(0.12+0.10+0.08)/0.49=0.612、F(0|0...)=0.612$
   • $P(x_2=1|x_{10}=0...)=(0.08+0.06+0.05)/0.49=0.388、F(1|0...)=1.000$
   • $u\sim U(0,1),u=0.65,u<F(1|0...)$ Make ：$x_{20}=1$
3. $x_3$ Of ( Conditions ) Marginal probability and cumulative distribution ：
   • $P(x_3=1|x_{10}{x}_{20}...)=0.08/0.19=0.421、F(1|0,1...)=0.421$
   • $P(x_3=2|x_{10}{x}_{20}...)=0.06/0.19=0.316、F(2|0,1...)=0.737$
   • $P(x_3=3|x_{10}{x}_{20}...)=0.05/0.19=0.263、F(3|0,1...)=1.000$
   • $u\sim U(0,1),u=0.55,u<F(1|0,1...)$ Make ：$x_{30}=2$
4. $(x_{10},x_{20},x_{30})=(0,1,2)$

Simulation generates multivariate random variables

import numpy as np
import matplotlib.pyplot as plt

def multivariateDiscrete(num=10):
    x1, x2, x3 = [], [], []
    for i in range(10000):
        u = np.random.uniform(0, 1)
        temp = []
        for k, v in {
    0:0.49,1:0.79,2:1.0}.items():
            if v > u:
                temp.append(k)
        x1.append(np.min(temp))

        u = np.random.uniform(0, 1)
        temp = []
        for k, v in {
    0: 0.612, 1:1.0}.items():
            if v > u:
                temp.append(k)
        x2.append(np.min(temp))

        u = np.random.uniform(0, 1)
        temp = []
        for k, v in {
    1:0.412, 2:0.737, 3:1.00}.items():
            if v > u:
                temp.append(k)
        x3.append(np.min(temp))
    return x1, x2, x3