Introduction to Probability, Second Edition, 2nd Edition

C

Table of distributions

Name	Param.	PMF or PDF	Mean	Variance
Bernoulli	$p$	$P\left(X=1\right)=p,P\left(X=0\right)=q$	$p$	$pq$
Binomial	$n,p$	$\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k{q}^{n-k},k\in \left\{0,1,\dots ,n\right\}$	$np$	$npq$
FS	$p$	$p{q}^{k-1},k\in \left\{1,2,\dots \right\}$	$1∕p$	$q∕p^2$
Geom	$p$	$p{q}^k,k\in \left\{0,1,2,\dots \right\}$	$q∕p$	$q∕p^2$
NBin	$r,p$	$\left(\genfrac{}{}{0ex}{}{r+k-1}{r-1}\right)p^r{q}^k,k\in \left\{0,1,2,\dots \right\}$	$rq∕p$	$rq∕p^2$
HGeom	$w,b,n$	$\frac{\left(\genfrac{}{}{0ex}{}{w}{k}\right)\left(\genfrac{}{}{0ex}{}{b}{n-k}\right)}{\left(\genfrac{}{}{0ex}{}{w+b}{n}\right)},k\in \left\{0,1,\dots ,n\right\}$	$\mu =\frac{nw}{w+b}$	$\left(\frac{w+b-n}{w+b-1}\right)\mu \left(1-\frac{\mu }{n}\right)$
NHGeom	$w,b,r$	$\frac{\left(\genfrac{}{}{0ex}{}{r+k-1}{r-1}\right)\left(\genfrac{}{}{0ex}{}{w+b-r-k}{w-r}\right)}{\left(\genfrac{}{}{0ex}{}{w+b}{w}\right)},k\in \left\{0,1,\dots ,b\right\}$	$\frac{rb}{w+1}$	$\frac{rb\left(w+b+1\right)\left(w-r+1\right)}{{\left(w+1\right)}^2\left(w+2\right)}$
Poisson	$\lambda$	$\frac{e^{-\lambda }{\lambda }^k}{k!},k\in \left\{0,1,2,\dots \right\}$	$\lambda$	$\lambda$
Uniform	$a<b$	$\frac{1}{b-a},x\in \left(a,b\right)$	$\frac{a+b}{2}$	$\frac{{\left(b-a\right)}^2}{12}$
Normal	$\mu ,{\sigma }^2$	$\frac{1}{\sigma \sqrt{2\pi }}{e}^{-{\left(x-\mu \right)}^2∕\left(2{\sigma }^2\right)}$	$\mu$	${\sigma }^2$
Log-Normal	$\mu ,{\sigma }^2$	$\frac{1}{x\sigma \sqrt{2\pi }}{e}^{-{\left(logx-\mu \right)}^2∕\left(2{\sigma }^2\right)},x>0$	$\theta =e^{\mu +{\sigma }^2∕2}$	${\theta }^2\left(e^{{\sigma }^2}-1\right)$
Expo	$\lambda$	$\lambda e^{-\lambda x},x>0$	$1∕\lambda$	$1∕{\lambda }^2$
Weibull	$\lambda ,\gamma$	$\gamma \lambda e^{-\lambda x^{\gamma }}{x}^{\gamma -1},x>0$	$\mu =\frac{\Gamma \left(1+1∕\gamma \right)}{{\lambda }^{1∕\gamma }}$	$\frac{\Gamma \left(1+2∕\gamma \right)}{{\lambda }^{2∕\gamma }}-{\mu }^2$
Gamma	$a,\lambda$	$\Gamma {\left(a\right)}^{-1}{\left(\lambda x\right)}^a{e}^{-\lambda x}{x}^{-1},x>0$	$a∕\lambda$	$a∕{\lambda }^2$
Beta	$a,b$	$\frac{\Gamma \left(a+b\right)}{\Gamma \left(a\right)\Gamma \left(b\right)}{x}^{a-1}{(1-}^{}$