## Uniform Distribution

• congruential method :
$x_{i+1}=(ax_i+c)\mod\; m, i=1, 2, ...,n-1$
• we should choose a, c, m carefully. e.g., $a = 25214903917, c = 11, m = 2^{48}$

• Uniformly distributed numbers on [0,1] can be obtained by : $u_i= {x_i \over {m-1} }, i=1,2,…,n$

• Uniformly distributed numbers on [a,b] can be obtained by : $u_i=a+ {x_i \over {m-1} } {(b-a)}, i=1,2,…,n$

## Bernoulli Distribution

• Generate u from uniform(0,1), if u < p , then return 1, else return 0.

## Binomial Distribution

• Generate $y_1,y_2,…,y_n$ from bernoulli distribution, return $y_1+y_2+…+y_n$.

## Cauchy Distribution $C(\alpha,\beta)$

• Generate u from uniform(0,1),return $x={ \alpha- {\beta \over tan(\pi u)}}$

## Empirical Distribution

• Generate u from uniform(0,1), let i be the integer part of $(n-1)u + 1$, return $a_i + [(n-1)u-i+1]{(a_{i+1}-a_i)}$

## Exponential Distribution

• Generate u from uniform(0,1), return $-\beta ln(u)$

## Erlang Distribution $ER(k,\beta)$

• Generate $y_1,y_2,…,y_k$ from exponential distribution $exp{(\beta \over k)}$, return $y_1 + y_2 +… +y_k$

## Gamma Distribution $G(\alpha,\beta)$

$x=0$
$repeat:$
$\quad generate \; v \; from \; exp(1)$
$\quad x=x+v$
$\quad \alpha=\alpha-1$
until $\alpha=1$
return $\beta x$

## Beta Distribution

• Generate $y_1$ from gamma distribution $G(\alpha,1)$
• Generate $y_2$ from gamma distribution $G(\beta,1)$
• $x= {y_1 \over {y_1 + y_2}}$
• return x

## Weibull Distribution $W(\alpha,\beta)$

• Generate v from $exp(1)$
• return $x={\beta v^{1 \over \alpha}}$

## Geometric Distribution GE(p)

• Generate u from uniform(0,1)
• return the integer part of $ln(u) \over ln{(1-p)}$

## Negative Binomial Distribution $NB(k,p)$

• Generate $y_1,y_2,…,y_k$ from geometric distribution GE(p)
• return $y_1+y_2+…+y_k$

## Logistic Distribution $L(a,b)$

• Generate u from uniform(0,1)
• return $a-{bln({1 \over u}-1)}$

## Normal Distribution $N(u,\sigma)$

• Generate $u_1$ from uniform(0,1)
• Generate $u_2$ from uniform(0,1)
• $z=[-2ln(u_1)]^{1 \over 2} sin(2\pi u_2)$
• return $u+\sigma z$

## Chi-Square Distribution

• Generate $z_i, i=1,2,…,k$ from $Normal(0,1)$
• return $z_1^2+z_2^2+…+z_k^2$

## F Distribution $F(k_1,k_2)$

• Generate $y_1$ from chi-square distribution $Chi(k_1)$
• Generate $y_2$ from chi-square distribution $Chi(k_2)$
• return $y1 / k1 \over y2/k2$

## Student Distribution $S(k)$

• Generate z from $N(0,1)$
• Generate y from $Chi(k)$
• return $z \over {\sqrt {y/k}}$

## Lognormal Distribution $LogN(u,\sigma^2)$

• Generate z from $N(0,1)$
• $x=u+\sigma z$
• return $exp(x)$

## Multinormal Distribution $N(u,\Sigma)$

• Generate an upper triangular matrix C such that $\Sigma=CC’$
• Generate $u_1,u_2,…,u_n$ from $N(0,1)$
• $x_k=u_k+ \sum_{i=1}^k c_{ki}u_i , (k=1,2,…,n)$
• return $x=(x_1,x_2,…x_n)$

## Triangular Distribution T(a,b,m)

• $c={(m-a)} \over {(b-a)}$
• Generate u from $uniform(0,1)$
• If $u < c$, then $y=\sqrt {cu}$
• else, $y=1-{\sqrt {(1-c)(1-u)}}$
• return $a+(b-a)y$

## Poisson distribution $P(\lambda)$

x=0
b=1
mark
generate u from $uniform(0,1)$
b=bu
if $b > e^\lambda$, then $x=x+1$ and goto mark

return x.