Table of Distributions
Distribution
Notation
Binomial
Bin(n, p)
Geometric
Geom(p)
Negative Binomial
NB(r, p)
HyperGeometric
HG(N, M, n)
Probability Function
Discrete Distributions
n x
pX (x) =
p (1 − p)n−x , x = 0, 1, . . . , n
x
pX (x) = p(1 − p)x−1 , x = 1, 2, . . .
x−1 r
pX (x) =
p (1 − p)x−r , x = r, r + 1, . . .
r − 1 M N −M
x
pX (x) =
n−x
N
n
Mean
Variance
Moment-Generating
Function
np
np(1 − p)
t
n
pe + (1 − p)
1
p
1−p
p2
r
p
r(1−p)
p2
nM
N
nM
N
N −M
N
N −n
pet
1 − (1 − p)et
r
pet
1 − (1 − p)et
N −1
x = max{0, n + M − N }, ..., min{n, M }
Poisson
Pois(µ)
pX (x) = e−µ
µx
, x = 0, 1, 2, . . .
x!
exp µ et − 1
µ
µ
a+b
2
(b−a)2
12
µ
σ2
β
β2
(1 − tβ)−1 , t < 1/β
αβ
αβ 2
(1 − tβ)−α , t < 1/β
d
2d
(1 − 2t)−d/2 , t < 1/2
α
α+β
αβ
(α+β)2 (α+β+1)
Continuous Distributions
Uniform
U(a, b)
Normal
N(µ, σ)
Exponential
Exp(β)
Gamma
G(α, β)
Chi-Squared
χ2d
Beta
Beta(α, β)
1
, a<x<b
b−a
1
(x − µ)2
√
fX (x) =
exp −
, −∞ < x < ∞
2σ 2
2πσ
1
fX (x) = exp (−x/β) , x > 0
β
1
fX (x) =
xα−1 e−x/β , x > 0
Γ(α)β α
d
1
−1 − x2
2
e , x>0
fX (x) =
x
Γ(d/2)2d/2
Γ(α + β) α−1
fX (x) =
x
(1 − x)β−1 , 0 < x < 1
Γ(α)Γ(β)
fX (x) =
Bivariate Normal
BN(µX , µY , σX , σY , ρ)
fX,Y (x, y) =
q=
1
2(1−ρ2 )
1
p
e−q ,
2πσX σY 1 − ρ2
x−µX
σX
2
x, y ∈ R
Y
X
Y
does not exist
2
σX
ρ σX σY
ρ σX σY
2
σY
µ
µY
x−µ
y−µY
y−µY 2
−2ρ σ X
+
σ
σ
X
etb − eta
t(b − a)
t2 σ 2
exp µt +
2
0
