Standard discrete and continuous distributions

Reference table

Distribution	Mean	Variance
Bernoulli(p)	p	pq
Binomial(n,p)	np	npq
Poisson$(\lambda)$	$\lambda$	$\lambda$
Geometric (trials, p)	1/p	$q/p^{2}$
Negative binomial(r,p)	r/p	rq$/p^{2}$
Hypergeometric	nK/N	nK$(N-K)(N-n)/(N^{2}(N-1))$
Uniform(a,b)	(a+b)/2	$(b-a)^{2}/12$
Exponential$(\lambda)$	$1/\lambda$	$1/\lambda^{2}$
Gamma$(k,\lambda)$	$k/\lambda$	$k/\lambda^{2}$
Beta(a,b)	a/(a+b)	ab$/((a+b)^{2}(a+b+1))$
$N(\mu,\sigma^{2})$	$\mu$	$\sigma^{2}$
$\chi^{2}_{n}$	n	2n
$t_\nu$	$0 (\nu>1)$	$\nu/(\nu-2) (\nu>2)$
Cauchy	—	—

Relationships to quote

• Sum of n i.i.d. Exp$(\lambda) =$ Gamma$(n, \lambda)$; Gamma(n/2, ½$) = \chi^{2}_{n}$.
• **min of independent Exp$(\lambda_{i})$ ~ Exp$(\sum\lambda_{i})**$ — the series-system fact.
• Sum of independent Poissons is Poisson; of independent normals is normal; of independent $\chi^{2}$ is $\chi^{2}$.
• Z ~ $N(0,1) \Rightarrow Z^{2}$ ~ $\chi^{2}_{1}. t = Z/\sqrt{\chi^{2}_{n}/n}; F = (\chi^{2}_{m}/m)/(\chi^{2}_{n}/n)$.
• Binomial → Poisson (n large, np fixed) → Normal (CLT).
• Memorylessness characterises the exponential (continuous) and geometric (discrete): P(X > s+t | X > s) = P(X > t).

Order statistics

For i.i.d. with CDF $F: P(X_{(n)} \le x) = F(x)^{n}$ and $P(X_{(1)} > x) = (1 - F(x))^{n}$. For Uniform$(0,1), X_{(k)}$ ~ Beta(k, n−k+1).

Key takeaways

• Exponential: memoryless, minimum stays exponential, sum becomes gamma. $- \chi^{2}, t$ and F are all built from normals — know the constructions.
• Uniform order statistics are Beta.