Generate random numbers from various probability distributions

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Words count in article 507

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.