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Probability Distributions

Module by: Don Johnson. E-mail the author

Table 1: Discrete Probability Distributions
Name Probability Mean Variance Relationships
Discrete Uniform {1N−M+1  if  M≤n≤N0  otherwise   1 N M 1 M n N 0 M+N2 M N 2 (N−M+2)⁢(N−M)12 N M 2 N M 12
Bernoulli Prn=0=1−p n 0 1 p , Prn=1=p n 1 p p p p⁢(1−p) p 1 p
Binomial Nn⁢pn⁢1−pN−n N n p n 1 p N n , n=0…N n 0 … N N⁢p N p N⁢p⁢(1−p) N p 1 p Sum of NN IID Bernoulli
Geometric (1−p)⁢pn 1 p p n , n≥0 n 0 p1−p p 1 p p1−p2 p 1 p 2
Negative Binomial n−1N−1⁢pN⁢1−pn−N n 1 N 1 p N 1 p n N , n≥N n N Np N p N⁢(1−p)p2 N 1 p p 2
Poisson λn⁢e−λn! λ n λ n , n≥0 n 0 λ λ λ λ
Hypergeometric an⁢bN−na+bN a n b N n a b N , n∈0…N n 0 … N ; 0≤n≤a+b 0 n a b ; 0≤N≤a+b 0 N a b N⁢aa+b N a a b N⁢a⁢b⁢(a+b−N)a+b2⁢(a+b−1) N a b a b N a b 2 a b 1
Logarithmic −pnn⁢log(1−p) p n n 1 p −p(1−p)⁢log(1−p) p 1 p 1 p (−p)⁢(p+log(1−p))(1−p)⁢logq p p 1 p 1 p q
Table 2: Distributions Related to the Gaussian
Name Probability Mean Variance Relationships
Gaussian (Normal) 12⁢π⁢σ2⁢e−(12⁢x−mσ2) 1 2 σ 2 1 2 x m σ 2 m m σ2 σ 2
Bivariate Gaussian 12⁢π⁢1−ρ2⁢ σ x ⁢ σ y ⁢e−(12×(1−ρ2)⁢(x− m x σ x 2−2⁢ρ⁢x− m x σ x ⁢y− m y σ y +y− m y σ y 2)) 1 2 1 ρ 2 σ x σ y 1 2 1 ρ 2 x m x σ x 2 2 ρ x m x σ x y m y σ y y m y σ y 2 Ex= m x x m x , Ey= m y y m y σ(x)2= σ x 2 x σ x 2 , σ(y)2= σ y 2 y σ y 2 , Ex⁢y= m x ⁢ m y +ρ⁢ σ x ⁢ρ⁢ σ y x y m x m y ρ σ x ρ σ y ρρ: correlation coefficient
Conditional Gaussian p⁢ x | y =12⁢π⁢1−ρ2⁢ σ x 2⁢e−x− m x −ρ⁢ σ x σ y ⁢(y− m y )22⁢ σ x 2×(1−ρ2) p x | y 1 2 1 ρ 2 σ x 2 x m x ρ σ x σ y y m y 2 2 σ x 2 1 ρ 2 m x +ρ⁢ σ x σ y ⁢(y− m y ) m x ρ σ x σ y y m y σ x 2⁢(1−ρ2) σ x 2 1 ρ 2
Generalized Gaussian 12⁢Γ⁢1+1r⁢A⁢r⁢e−|x−mA⁢r|r 1 2 Γ 1 1 r A r x m A r r m m σ2 σ 2 A⁢r=σ2⁢Γ⁢1rΓ⁢3r A r σ 2 Γ 1 r Γ 3 r
Chi-Squared ( χ ν 2 χ ν 2 ) 12ν2⁢Γ⁢ν2⁢xν2−1⁢e−x2 1 2 ν 2 Γ ν 2 x ν 2 1 x 2 , 0≤x 0 x ν ν 2⁢ν 2 ν χ ν 2=∑ i =1ν x i 2 χ ν 2 i 1 ν x i 2 , x i x i IID 𝒩01 0 1
Noncentral Chi-Squared ( χ ν ′ 2λ χ ν ′ λ 2 ) 12⁢xλν−24⁢ I ( ν − 2 ) / 2 ⁢(λ⁢x)⁢e−(12⁢(λ+x)) 1 2 x λ ν 2 4 I ( ν − 2 ) / 2 λ x 1 2 λ x ν+λ ν λ 2⁢(ν+2⁢λ) 2 ν 2 λ χ ν ′ 2=∑ i =1ν x i 2 χ ν ′ 2 i 1 ν x i 2 , x i x i IID 𝒩 m i 1 m i 1 , λ=∑ i =1ν m i 2 λ i 1 ν m i 2
Student's t Γ⁢ν+12ν⁢π⁢Γ⁢ν2⁢1+x2ν−ν+12 Γ ν 1 2 ν Γ ν 2 1 x 2 ν ν 1 2 0 0 νν−2 ν ν 2 , 2<ν 2 ν
Beta β m , n β m , n Γ⁢m+n2Γ⁢m2⁢Γ⁢n2⁢xm2−1⁢1−xn2−1 Γ m n 2 Γ m 2 Γ n 2 x m 2 1 1 x n 2 1 , 0<x<1 0 x 1 , 0<m∧n 0 m n mm+n m m n 2⁢m⁢nm+n2⁢(m+n+2) 2 m n m n 2 m n 2 β m , n = χ m 2 χ m 2+ χ n 2 β m , n χ m 2 χ m 2 χ n 2
F Distribution Γ⁢m+n2Γ⁢m2⁢Γ⁢n2⁢mnm2⁢xm−221+mn⁢xm+n2 Γ m n 2 Γ m 2 Γ n 2 m n m 2 x m 2 2 1 m n x m n 2 , 0≤x 0 x , 1≤m∧n 1 m n nn−2 n n 2 , n>2 n 2 2⁢n2⁢(m+n−2)m⁢n−22⁢(n−4) 2 n 2 m n 2 m n 2 2 n 4 , n>4 n 4 F m , n = χ m 2m χ n 2n F m , n χ m 2 m χ n 2 n
Non-central F [ F m , n ′ ⁢λ F m , n ′ λ ] ∑ k =0∞λ2kk!⁢e−λ2⁢ p β m 2 + k , n 2 ⁢m⁢xm⁢x+n k 0 λ 2 k k λ 2 p β m 2 + k , n 2 m x m x n nn−2 n n 2 , n>2 n 2 2⁢nm2⁢m+λ2+(m+2⁢λ)⁢(n−2)n−22⁢(n−4) 2 n m 2 m λ 2 m 2 λ n 2 n 2 2 n 4 , n>4 n 4 F m , n ′ ⁢λ= χ m ′ 2λm χ n 2λn F m , n ′ λ χ m ′ λ 2 m χ n λ 2 n
Wishart W M ⁢NK W M N K detwN−M−122N⁢M2⁢ Γ M ⁢N2⁢detKK2⁢e−tr⁢K-1⁢w2 w N M 1 2 2 N M 2 Γ M N 2 K K 2 tr K w 2 N⁢K N K cov⁢Wi,j⁢Wk,l=N⁢(Ki,k⁢Kj,l+Ki,l⁢Kj,k) cov W i j W k l N K i k K j l K i l K j k W M ⁢NK=∑ n =1N x n ⁢ x n H W M N K n 1 N x n x n , x n ∼𝒩0K x n 0 K , dim⁢x=M dim x M , Γ M ⁢N2=πM⁢(M−1)4⁢∏ m =0M−1Γ⁢N2−m2 Γ M N 2 π M M 1 4 m 0 M 1 Γ N 2 m 2
Table 3: Non-Gaussian Distributions
Name Probability Mean Variance Relationships
Uniform 1b−a 1 b a , a≤x≤b a x b a+b2 a b 2 b−a212 b a 2 12
Triangular {2⁢xa  if  0≤x≤a2×(1−x)1−a  if  a≤x≤1 2 x a 0 x a 2 1 x 1 a a x 1 1+a3 1 a 3 1−a+a218 1 a a 2 18
Exponential λ⁢e−(λ⁢x) λ λ x , 0≤x 0 x 1λ 1 λ 1λ2 1 λ 2
Lognormal 12⁢π⁢σ2⁢x2⁢e−(12⁢logx−mσ2) 1 2 σ 2 x 2 1 2 x m σ 2 , 0<x 0 x em+σ22 m σ 2 2 e2⁢m⁢(e2⁢σ2−eσ2) 2 m 2 σ 2 σ 2
Maxwell 2π⁢a32⁢x2⁢e−a⁢x22 2 a 3 2 x 2 a x 2 2 , 0<x 0 x 8π⁢a 8 a (3−8π)⁢a-1 3 8 a
Laplacian 12⁢σ2⁢e−|x−m|σ22 1 2 σ 2 x m σ 2 2 m m σ2 σ 2
Gamma baΓ⁢a⁢xa−1⁢e−(b⁢x) b a Γ a x a 1 b x , 0<x 0 x , 0<a∧b 0 a b ab a b ab2 a b 2
Rayleigh 2⁢a⁢x⁢e−(a⁢x2) 2 a x a x 2 , 0≤x 0 x π4⁢a 4 a 1a⁢(1−π4) 1 a 1 4
Weibull a⁢b⁢xb−1⁢e−(a⁢xb) a b x b 1 a x b , 0<x 0 x , 0<a∧b 0 a b 1a1b⁢Γ⁢1+1b 1 a 1 b Γ 1 1 b a−2b⁢(Γ⁢1+2b−Γ21+1b) a 2 b Γ 1 2 b Γ 1 1 b 2
Arc-Sine 1π⁢x⁢(1−x) 1 x 1 x , 0<x<1 0 x 1 12 1 2 18 1 8
Circular Normal ea⁢cosx−m2⁢π⁢ I 0 ⁢a a x m 2 I 0 a , −π<x≤π x m m
Cauchy aπx−m2+a2 a x m 2 a 2 m m (from symmetry arguments) ∞
Logistic e−x−maa⁢1+e−x−ma2 x m a a 1 x m a 2 , 0<a 0 a m m a2⁢π23 a 2 2 3
Gumbel e−x−maa⁢e−e−x−ma x m a a x m a , 0<a 0 a m+a⁢γ m a γ a2⁢π26 a 2 2 6
Pareto a⁢bax1−a a b a x 1 a , 0<a 0 a ; 0<b≤x 0 b x a⁢ba−1 a b a 1 , a>1 a 1 a⁢b2(a−2)⁢a−12 a b 2 a 2 a 1 2 , a>2 a 2

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