Introduction to Probability

Chapter C

Table of distributions

Name	Param.	PMF or PDF	Mean	Variance
Bernoulli	p	P (X = 1) = p, P(X = 0) = q	p	pq
Binomial	n,p	${(}_k^n)p^k{q}^{n-k},\text{for}k\in \{0,1,...,n\}$	np	npq
FS	p	pqk−1, for k ∈{1,2,...}	1/p	q/p2
Geom	p	pqk , for k ∈{0,1,2,...}	q/p	q/p2
NBinom	r,p	${(}_{r-1}^{r+n-1})p^r{q}^n,n\in \{0,1,2,...\}$	rq/p	rq/p2
HGeom	w, b, n	$\frac{{(}_k^w){(}_{n-k}^b)}{{(}_n^{w+b})},\text{for}k\in \{0,1,...,n\}$	$\mu =\frac{nw}{w+b}$	$\left(\frac{w+b-n}{w+b-1}\right)n\frac{\mu }{n}\left(1-\frac{\mu }{n}\right)$
Poisson	λ	$\frac{e^{-\lambda }{\lambda }^k}{k!},\text{for}k\in \{0,1,2,...\}$	λ	λ
Uniform	a < b	$\frac{1}{b-a},\text{for}x\in (a,b)$	$\frac{a+b}{2}$	$\frac{{\left(b-a\right)}^2}{12}$
Normal	μ, σ2	$\frac{1}{\sigma \sqrt{2\pi }}{e}^{-{(x-\mu )}^2/(2{\sigma }^2)}$	μ	σ2
Log-Normal	μ, σ2	$\frac{1}{x\sigma \sqrt{2}}$