MAT2013 Mathematical Statistics: Formula Sheet
Discrete Distributions
Distribution
Mass function
Ranges of variables
Mean and Variance
Uniform (N )
1
N
N = 1, 2, . . .
N +1
2
x = 1, 2, . . . , N
N 2 −1
12
0 ≤ π ≤ 1,
π
r = 0, 1
π(1 − π)
0 ≤ π ≤ 1, n = 1, 2, . . .
nπ
r = 0, 1, . . . n
nπ(1 − π)
λ>0
λ
r = 0, 1, 2, . . .
λ
0<π≤1
1
π
n = 1, 2, . . .
(1−π)
π2
0 < π ≤ 1, k > 0
k
π
x = k, k + 1, . . .
k(1−π)
π2
n1 , n2 , m = 1, 2, . . .
n1 m
n1 +n2
x = 0, 1, ..., min(n1 , m)
n1 n2 m(n1 +n2 −m)
(n1 +n2 )2 (n1 +n2 −1)
0 ≤ πj ≤ 1, m = 1, 2, . . .
mπj
π r (1 − π)1−r
Bernoulli (π)
Binomial (n, π)
n
r
π r (1 − π)n−r
e−λ λr
r!
Poisson (λ)
(1 − π)n−1 π
Geometric (π)
Negative Binomial (k, π)
x−1
k−1
n2
(nx1 )(m−x
)
n1 +n2
( m )
Hypergeometric (n1 , n2 , m)
Multinomial (k, m, π)
π k (1 − π)x−k
m
x
π1x1 π2x2 · · · πkxk
xj = 0, 1, . . . , m,
Pk
j=1
xj = m
m(δjk πj − πj πk )
MAT2013 Mathematical Statistics: Formula Sheet
Continuous Distributions
Distribution
Density function
Ranges of variables
Mean and Variance
1
β−α
−∞ < α < β < ∞
α+β
2
a<x<b
(β−α)2
12
λ>0
1
λ
x>0
1
λ2
−∞ < µ < ∞, σ > 0
µ
−∞ < x < ∞
σ2
α > 0, β > 0
α
β
x>0
α
β2
α > 0, β > 0
α
α+β
0<x<1
αβ
(α+β)2 (α+β+1)
−∞ < µj < ∞, Σ > 0
µj
−∞ < xj < ∞
Σjk
ν = 1, 2, . . .
ν
u>0
2ν
ν>0
µ = 0, ν > 1
Uniform (α, β)
λe−λx
Exponential (λ)
√1 e−
σ 2π
Normal (µ, σ 2 )
(x−µ)2
2σ 2
β α xα−1 exp(−βx)
Γ(α)
Gamma (α, β)
st Γ(α) = (α − 1)!
xα−1 (1−x)β−1
B(α,β)
Beta (α, β)
Γ(α)Γ(β)
Γ(α+β)
st B(α, β) =
Multivariate normal
for α ∈ N
1
|2πΣ|−1/2 e− 2 (x−µ)
T Σ−1 (x−µ)
(µ, Σ)
t(ν)
1
2 2 ν Γ( 12 ν)
1
1
2
1
B( 21 , ν2 )ν 2 (1+ tν ) 2 (ν+1)
F (ν1 , ν2 )
1
1
u 2 ν−1 e− 2 u
χ2 (ν)
B(
ν1
ν2
1 ν1
2
−∞ < t < ∞
ν
,
ν−2
ν>2
ν1 , ν2 = 1, 2, . . .
ν2
,
ν2 −2
ν2 > 2
x>0
2ν22 (ν1 +ν2 −2)
,
ν1 (ν2 −2)2 (ν2 −4)
1
x 2 ν1 −1
ν1 ν2
ν x 1
, )(1+ ν1 ) 2 (ν1 +ν2 )
2 2
2
ν2 > 4
Some Useful formulae
1. The pgf of Poisson(λ) is e−λ(1−z) .
2. The mgf of N(µ, σ 2 ) is exp(µz + σ 2 z 2 /2).
3. The mgf of gamma(α, β) is (1 − βz )−α .
4. The Cauchy Schwartz inequality states that for any two random variables X, Y ,
[E(XY )]2 ≤ E(X 2 )E(Y 2 ).
5. Chebyshev’s inequality states that for a random variable Y with mean
µ and finite variance σ 2 , P (|Y − µ| ≥ a) ≤ σ 2 /a2 , for all a > 0.
6. Exponential series formula: ex =
xn
n=0 n! .
P∞
7. Where needed you may use that for λ > 0,
R∞
0
ue−λu = 1/λ2 .