MCS: discrete random variable

Discrete Arbitrary

$x$ Is a discrete variable , It represents a set of values ,$x_i$ For the th in the set $i$ Number ,($i\sim 1\to N$), A specific value in a collection $x_i$ The probability of is written as ：$P(x_i)=P(x=x_i)$, therefore $P(x_1),...P(x_N)$ Defined variables $x$ Probability distribution of .

$$\sum _{i}P(x_i)=1$$

Variable $x$ Expectation and variance of ：

$$E(x)=\sum _i{x}_{i}P(x_i)$$

$$V(x)=E(x^2)-E(x{)}^2$$

$$E(x^2)=\sum _i{x}_i^{2}P(x_i)$$

CDF:

$$F(x_i)=P(x<=x_i)$$

Generate random variables that obey any probability distribution

1. For each $x_i$, find $F(x_i),i\sim 1\to N$
2. Generate a random continuous uniform $u\sim U(0,1)$
3. Locate the smallest $x_i$ where $u<F(x_i)$
4. $ x = x_i$
5. Return $x$

example ： hypothesis $x$ Is discrete , Obey the following probability distribution and cumulative distribution function ：

x	P(x)	F(x)
0	0.4	0.4
1	0.3	0.7
2	0.2	0.9
3	0.1	1.0

1. Generate a uniformly distributed random variable ：$u\sim U(0,1),u=0.58$
2. $u<F(1)=0.7$
3. $x=1$