이산형 분포

베르누이 분포 (Bernoulli distribution)

$p(x)=p^x(1-p)^{1-x}$,    $x=0,1$,   $0<p<1$

$\mu=p$

$\sigma=p(1-p$

$m(t)=[(1-p)+pe^t]$,  $\infty<t<\infty$

 

이항 분포 (Binomial)

$p(x)= {n \choose x} p^x(1-p)^{n-x}$,    $x=0,1,2,\cdots,n$,   $0<p<1$

$\mu=np$

$\sigma^2=np(1-p)$

$m(t)=[(1-p)+pe^t]^n$, $-\infty<t<\infty$

 

기하 분포 (Geometric)

$p(x)=p(1-p)^x$,    $x=0,1,2,\cdots$,   $0<p<1$

$\mu=\frac{p}{q}$

$\sigma^2=\frac{1-p}{p^2}$

$m(t)=p[1-(1-p)e^t]^{-1}$, $t<-log(1-p)$

 

초기하 분포 (Hypergeometric)

$p(x)=\frac{{N-D \choose n-x}{D \choose x}}{N \choose n}, x=0,1,2,\cdots,n$

$\mu=n\frac{D}{N}$

$\sigma^2=n\frac{D}{N}\frac{N-D}{N}\frac{N-n}{N-1}$

 

음이항 분포 (Negative Binomial)

$p(x)={x+r-1 \choose r-1}p^r(1-p)^x, x=0,1,2,\cdots, 0<p<1, r=1,2,\cdots$

$\mu=\frac{rp}{q}$

$\sigma^2=\frac{r(1-p)}{p^2}$

$m(t)=p^r[1-(1-p)e^t]^{-r}, t<-log(1-p)$

 

푸아송 분포 (Poisson)

$p(x)=\frac{e^{-m}m^x}{x!}, x=0,1,2,\cdots, m>0$

$\mu=m$

$\sigma^2=m$

$m(t)=exp[m(e^t-1)], -\infty<t<\infty$

 

다항 분포 (Multinomial)

$p(x_1,x_2,\cdots,x_{k-1}=\frac{n!}{x_1!\cdots x_{k-1}!x_k!} p_1^{x_1}\cdots p_{k-1}^{x_{k-1}}p_k^{x_k}, x_k=n-x_1-\cdots-x_{k-1}$

$E(X_i)=np_i$

$Var(X_i)=np_i(1-p_i)$

$Cov(X_i,X_j)=-np_ip_j, i!=j$

$m(t_1,t_2,\cdots,t_n)=(\sum_{i=1}^{k} p_ie^{t_i})^n$