MCS: discrete random variable - uniform distribution

Discrete Uniform

Variable $x$ Obey uniform discrete distribution , There are two parameters $a,b$,$x$ Accept $a\sim b$ All integers in the range ,$P(x)$:

$$P(x)=\frac{1}{b-a+1},x\sim a\to b$$

CDF:

$$F(x)=\frac{x-a+1}{b-a+1},x\sim a\to b$$

Expectation and variance ：

$$E(x)=(a+b)/2$$

$$V(x)=\frac{(b-a+1{)}^2-1}{12}$$

Generate random discrete uniform variables

1. Generate a random continuous uniform $u\sim U(0,1)$
2. $x=ceiling[(a-1)+u(b-a+1)]$
3. Return $x$

example ： Suppose a random variable $x$, Obey uniform discrete distribution , The value is $10\sim 20$ All integers between , The random variable is generated by simulation ：

1. Generate a uniform continuous random variable ： $u=0.71$
2. Calculation $10\sim 20$ Between discrete random variables ：$x=ceiling[(10-1)+0.71\times (20-10+1)]=ceiling[16.81]=17$
3. $x=17$