Table of Common Distributions

Table of Common Distributions taken from Statistical Inference by Casella and Berger

Discrete Distrbutions distribution

pmf

mean

variance

mgf/moment

Bernoulli(p)

px (1 − p)1−x ; x = 0, 1; p ∈ (0, 1)

p

p(1 − p)

(1 − p) + pet

Beta-binomial(n, α, β)

Γ(α+β) Γ(x+α)Γ(n−x+β) (nx ) Γ(α)Γ(β) Γ(α+β+n)

nα α+β

nαβ (α+β)2

Notes: If X|P is binomial (n, P ) and P is beta(α, β), then X is beta-binomial(n, α, β). Binomial(n, p)

( nx )px (1 − p)n−x ; x = 1, . . . , n

np

np(1 − p)

Discrete Uniform(N )

1 N;

N +1 2

(N +1)(N −1) 12

[(1 − p) + pet ]n PN it 1 i=1 e N

Geometric(p)

p(1 − p)x−1 ; p ∈ (0, 1)

1 p

1−p p2

pet 1−(1−p)et

x = 1, . . . , N

Note: Y = X − 1 is negative binomial(1, p). The distribution is memoryless: P (X > s|X > t) = P (X > s − t). Hypergeometric(N, M, K)

N −M (M x )( K−x ) ; N ( K)

x = 1, . . . , K

KM N

KM (N −M )(N −k) N N (N −1)

r(1−p) p

r(1−p) p2

λ

λ

M − (N − K) ≤ x ≤ M ; N, M, K > 0 Negative Binomial(r, p)

)pr (1 − p)x ; p ∈ (0, 1) (r+x−1 x

? ³

p 1−(1−p)et

r y−r ( y−1 ; Y =X +r r−1 )p (1 − p)

Poisson(λ)

e−λ λx x! ;

λ≥0

Notes: If Y is gamma(α, β), X is Poisson( βx ), and α is an integer, then P (X ≥ α) = P (Y ≤ y).

1

t

eλ(e

−1)

´r

Continuous Distributions distribution

pdf

Beta(α, β)

Γ(α+β) α−1 (1 Γ(α)Γ(β) x

Cauchy(θ, σ)

1 1 πσ 1+( x−θ )2 ; σ

− x)β−1 ; x ∈ (0, 1), α, β > 0

σ>0

mean

variance

α α+β

αβ (α+β)2 (α+β+1)

mgf/moment P∞ ³Qk−1 1 + k=1 r=0

does not exist

does not exist

does not exist

Notes: Special case of Students’s t with 1 degree of freedom. Also, if X, Y are iid N (0, 1), χ2p Notes: Gamma( p2 , 2). Double Exponential(µ, σ) Exponential(θ)

p

x 2 −1 e− 2 ; x > 0, p ∈ N

1 p 2 Γ( p 2 )2

x

1 − |x−µ| σ ; 2σ e 1 −x θ ; x ≥ θe

σ>0 0, θ > 0

q

1

p

2p

µ

2σ 2

θ

θ2

X Y

is Cauchy

³

1 1−2t

´ p2

eµt 1−(σt)2 1 1−θt , t

X Notes: Gamma(1, θ). Memoryless. Y = X γ is Weibull. Y = 2X β is Rayleigh. Y = α − γ log β is Gumbel. ³ ´ ν21 ν1 −2 ν +ν Γ( 1 2 2 ) ν1 ν2 +ν2 −2 x 2 2 ; x>0 2( ν2ν−2 )2 νν11 (ν , ν2 > 4 Fν1 ,ν2 ν1 ν2 ¡ ν1 ¢ ν1 +ν 2 ν2 −2 , ν2 > 2 Γ( 2 )Γ( 2 ) ν2 2 −4) 2

, t
0

x−µ β

1

Notes: The cdf is F (x|µ, β) =

−

1+e

x−µ β

³

αβ

1 X

π2 β 2 3

µ

1 1−βt

´α , t
0

βα β−1 ,

tν

Γ( ν+1 2 ) Γ( ν2 )

√1 νπ

0, ν > 1

ν ν−2 , ν

b+a 2

(b−a)2 12

ebt −eat (b−a)t

h i 2 β γ Γ(1 + γ2 ) − Γ2 (1 + γ1 )

EX n = β γ Γ(1 + nγ )

Notes: t2ν = F1,ν .

(x−µ)2 2σ 2

; x > 0, σ > 0

; σ>0

1 2 (1+ xν

ν+1 ) 2

1 Uniform(a, b) b−a , a ≤ x ≤ b Notes: If a = 0, b = 1, this is special case of beta (α = β = 1). xγ

γ γ−1 − β e ; x > 0, γ, β > 0 Weibull(γ, β) βx Notes: The mgf only exists for γ ≥ 1.

eµ+

2

e2(µ+σ ) − e2µ+σ

2

σ2

µ

1

β>1

β γ Γ(1 + γ1 )

2

EX n = enµ+ eµt+

βα2 (β−1)2 (β−2) ,

β>2

>2

σ 2 t2 2

does not exist EX n =

n n Γ( ν+1 2 )Γ(ν− 2 ) √ ν2, πΓ( ν2 )

n

n even

ν2 2