Probability Distributions Cheat Sheet

Discrete Distributions

Distribution	PMF / Parameters	Mean	Variance	MGF
Bernoulli(p)	P(X=1) = p, P(X=0) = 1-p	p	p(1-p)	(1-p) + pe^t
Binomial(n, p)	C(n,k) p^k (1-p)^(n-k)	np	np(1-p)	((1-p) + pe^t)^n
Poisson(lambda)	e^(-lambda) lambda^k / k!	lambda	lambda	e^(lambda(e^t - 1))
Geometric(p)	(1-p)^(k-1) p, k=1,2,...	1/p	(1-p)/p^2	pe^t / (1 - (1-p)e^t)
NegBin(r, p)	C(k-1, r-1) p^r (1-p)^(k-r)	r/p	r(1-p)/p^2	(pe^t / (1-(1-p)e^t))^r

Continuous Distributions

Distribution	PDF	Mean	Variance
Uniform(a, b)	1/(b-a) for a < x < b	(a+b)/2	(b-a)^2/12
Exponential(theta)	(1/theta) e^(-x/theta)	theta	theta^2
Normal(mu, sigma^2)	(1/(sigma sqrt(2pi))) e^(-(x-mu)^2/(2sigma^2))	mu	sigma^2
Gamma(alpha, theta)	x^(alpha-1) e^(-x/theta) / (theta^alpha Gamma(alpha))	alpha * theta	alpha * theta^2
Beta(a, b)	x^(a-1)(1-x)^(b-1) / B(a,b)	a/(a+b)	ab/((a+b)^2(a+b+1))
Pareto(alpha, theta)	alpha theta^alpha / (x+theta)^(alpha+1)	theta/(alpha-1)	alpha theta^2 / ((alpha-1)^2(alpha-2))
Lognormal(mu, sigma^2)	(1/(x sigma sqrt(2pi))) e^(-(ln x - mu)^2/(2sigma^2))	e^(mu + sigma^2/2)	(e^(sigma^2) - 1) e^(2mu + sigma^2)
Weibull(tau, theta)	(tau/theta)(x/theta)^(tau-1) e^(-(x/theta)^tau)	theta Gamma(1 + 1/tau)	theta^2[Gamma(1+2/tau) - (Gamma(1+1/tau))^2]

Key Relationships

• Exponential is Gamma(1, theta)
• Chi-squared(n) is Gamma(n/2, 2)
• Sum of n independent Exp(theta) is Gamma(n, theta)
• Sum of independent Poisson(lambda_i) is Poisson(sum lambda_i)
• Binomial(n, p) approaches Poisson(np) when n is large, p is small