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	${(}_k^n)p^k{q}^{n−k},\text{for}k∈\{0,1,...,n\}$	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	${(}_{r−1}^{r+n−1})p^r{q}^n,n∈\{0,1,2,...\}$	rq/p	rq/p2
HGeom	w, b, n	$\frac{{(}_k^w){(}_{n−k}^b)}{{(}_n^{w+b})},\text{for}k∈\{0,1,...,n\}$	$\mathrm{μ}=\frac{nw}{w+b}$	$\left(\frac{w+b−n}{w+b−1}\right)n\frac{\mathrm{μ}}{n}\left(1−\frac{\mathrm{μ}}{n}\right)$
Poisson	λ	$\frac{e^{−\mathrm{λ}}{\mathrm{λ}}^k}{k!},\text{for}k∈\{0,1,2,...\}$	λ	λ
Uniform	a < b	$\frac{1}{b−a},\text{for}x∈(a,b)$	$\frac{a+b}{2}$	$\frac{{\left(b−a\right)}^2}{12}$
Normal	μ, σ2	$\frac{1}{\mathrm{σ}\sqrt{2\mathrm{π}}}{e}^{−{(x−\mathrm{μ})}^{}}$