Some Absolutely Continuous Distributions
Name
Notation
p.d.f.
range
m.g.f.
mean
variance
Uniform
U (a, b)
1
b−a
a≤x≤b
ebθ −eaθ
(b−a)θ
a+b
2
(b−a)2
12
Normal
N (µ, σ 2 )
µ
σ2
Gamma
Γ(t, λ)
λt t−1 −λx
e
Γ(t) x
0<x<∞,
t,λ>0
¢t
λ
λ−θ
t
λ
t
λ2
Exponential
Exp(λ)
λe−λx
0<x<∞,
λ>0
λ
λ−θ
1
λ
1
λ2
Chi-square
χ2n
1
xn/2−1 e−x/2
2n/2 Γ(n/2)
0<x<∞,
n>0 integer
(1 − 2θ)−n/2
n
2n
0<x<1,
α1 ,α2 >0
not useful
α1
α1 +α2
α1 α2
(α1 +α2 )2 (α1 +α2 +1)
−∞<x<∞,
λ>0
doesn’t exist
but char. fn.
is eiµθ−λ|θ|
√ 1
2πσ 2
Cauchy
Multivariate
Normal
(x−µ)2 ª
2σ 2
Γ(α1 +α2 ) α1 −1
(1
Γ(α1 )Γ(α2 ) x
β(α1 , α2 )
Beta
©
exp −
− x)α2 −1
λ
π(λ2 +(x−µ)2 )
–
exp
N (µ, Σ)
−∞ < x < ∞
©
ª
− 12 (x−µ)T Σ−1 (x−µ)
(2π)d/2 det(Σ)1/2
x ∈ Rd
1
eµθ+ 2 σ
¡
T
eµ
2 2
θ
1
θ+ 2 θ T Σθ
no moments exist
µ
covariance
matrix Σ
Some Discrete Distributions
(All distributions below have integer-valued ranges)
Name
Notation
Binomial
Bin(n, p)
Poisson
Pois(λ)
prob. fn.
range
m.g.f.
mean
variance
px (1 − p)n−x
0≤x≤n
0<p<1
[(1 − p) + peθ ]n
np
np(1 − p)
λ
λ
¡n¢
x
e−λ λx
x!
a
x
b
n−x
a+b
n
θ
0≤x<∞
λ>0 real
eλ(e
−1)
0≤x≤n
a,b>0 integers
not useful
na
a+b
nab(a+b−n)
(a+b)2 (a+b−1)
Nbin(k, p)
( )( )
( )
¡x−1¢ k
x−k
k−1 p (1 − p)
k≤x<∞,
0<p<1
pk [e−θ − (1 − p)]−k
k/p
k(1 − p)/p2
Geo(p)
p(1 − p)x−1
1≤x<∞
0<p<1
p[e−θ − (1 − p)]−1
1/p
(1 − p)/p2
Hypergeometric
Hgeo(n, a, b)
Negative
binomial
Geometric
Multinomial: multivariate extension of binomial
Prob. fn. is given by
f (x1 , x2 , . . . , xk ) =
n!
xk+1
px1 px2 . . . pxkk pk+1
,
x! !x2 ! . . . xk !xk+1 ! 1 2
0 ≤ xi ≤ n, 0 < pi < 1 ∀i = 1, 2, . . . , n + 1
where p1 + p2 + · · · + pk+1 = 1 and x1 + x2 + · · · + xk+1 = n.
The m.g.f. is given by
[p1 eθ1 + p2 eθ2 + · · · + pk eθk + pk+1 ]n .
Moments: E(Xi ) = npi , Var(Xi ) = npi (1 − pi ), cov(Xi , Xj ) = −npi pj , i, j = 1, 2, . . . , n + 1.