X~Exp(λ)

pmf=λe^-λx 

for x>=0

cdf= 1-e^-λx

for x>=0

X~Gamma(a)

pdf= (λ^a/Γ(a))x^a-1e^-λx

X~Beta(a,b)

pmf= x^a−1(1 − x)^b−1/B(a,b)

X~N(μ,σ²)

pmf= (1/σ√2π)e^{(-(x-μ)^2/2σ^2)}

cdf= Φ(x-μ/σ)

X~Unif(a,b)

pmf= 1/(b-a)

cdf= (x-a)/(b-a) for a<x<b

=1 x>b

X~Cauchy

pmf= 1/π(1-x²)

cdf= .5+ (1/π)arctan x

E(X)=1/λ

V(X)=1/λ²

E(X)= a/λ

V(X)= a/λ²

E(X)= a/(a+b)

V(X)= ab/(a+b)²(a+b+1)

E(X)=μ

V(X)=σ²

E(X)=(a+b)/2

V(X)=(b-a)²/12