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<
Γ(
1+( ν )x
´
tk
k!
1
2
1
θ
<
EX n =
α+r
α+β+r
ν1 +2n
ν −2n
)Γ( 2 2 )
2
ν
ν
Γ( 21 )Γ( 22 )
³
ν2
ν1
´n
, n<
2
Notes: Fν1 ,ν2 =
χ2ν1 /ν1
χ2ν2 /ν2 ,
2
where the χ s are independent. F1,ν = t2ν .
x
1
α−1 − β
e ;
Γ(α)β α x
αβ 2
q
Notes: Some special cases are exponential (α = 1) and χ2 (α = p2 , β = 2). If α = 23 , Y = X
β is Maxwell. Y =
Gamma(α, β)
Logistic(µ, β)
1
β
h
−
e
x−µ
β
−
1+e
x > 0, α, β > 0
i2 ; β > 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<
1
β
is inverted gamma.
eµt Γ(1 + βt), |t| <
1
β
.
(log x−µ)2
2σ 2
σ2
2
n2 σ 2
2
Lognormal(µ, σ 2 )
1 −
√1
e
2πσ x
Normal(µ, σ 2 )
√ 1 e−
2πσ
Pareto(α, β)
βαβ
;
xβ+1
x > α, α, β > 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