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Probability Distributions

Module by: Don Johnson. E-mail the author

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Table 1: Discrete Probability Distributions
Name Probability Mean Variance Relationships
Discrete Uniform 1N−M+1ifM≤n≤N0otherwise 1 N M 1 M n N 0 M+N2 M N 2 N−M+2N−M12 N M 2 N M 12
Bernoulli Prn=0=1−p n 0 1 p , Prn=1=p n 1 p p p p1−p p 1 p
Binomial Nnpn1−pN−n N n p n 1 p N n , n=0…N n 0 … N Np N p Np1−p N p 1 p Sum of NN IID Bernoulli
Geometric 1−ppn 1 p p n , n≥0 n 0 p1−p p 1 p p1−p2 p 1 p 2
Negative Binomial n−1N−1pN1−pn−N n 1 N 1 p N 1 p n N , n≥N n N Np N p N1−pp2 N 1 p p 2
Poisson λnⅇ-λn! λ n λ n , n≥0 n 0 λ λ λ λ
Hypergeometric anbN−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 Naa+b N a a b Naba+b−Na+b2a+b−1 N a b a b N a b 2 a b 1
Logarithmic -pnnlog1−p p n n 1 p -p1−plog1−p p 1 p 1 p -pp+log1−p1−plogq p p 1 p 1 p q
Table 2: Distributions Related to the Gaussian
Name Probability Mean Variance Relationships
Gaussian (Normal) 12πσ2ⅇ-12x−mσ2 1 2 σ 2 1 2 x m σ 2 m m σ2 σ 2
Bivariate Gaussian 12π1−ρ2 σ x σ y ⅇ-121−ρ2x− m x σ x 2−2ρx− 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 σx2= σ x 2 x σ x 2 , σy2= σ y 2 y σ y 2 , Exy= 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 =12π1−ρ2 σ x 2ⅇ-x− m x −ρ σ x σ y y− m y 22 σ x 21−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 21−ρ2 σ x 2 1 ρ 2
Generalized Gaussian 12Γ1+1rArⅇ-|x−mAr|r 1 2 Γ 1 1 r A r x m A r r m m σ2 σ 2 Ar=σ2Γ1rΓ3r A r σ 2 Γ 1 r Γ 3 r
Chi-Squared ( χ ν 2 χ ν 2 ) 12ν2Γν2xν2−1ⅇ-x2 1 2 ν 2 Γ ν 2 x ν 2 1 x 2 , 0≤x 0 x ν ν 2ν 2 ν χ ν 2=∑i=1ν x i 2 χ ν 2 i 1 ν x i 2 , x i x i IID 01 0 1
Noncentral Chi-Squared ( χ ν ′ 2λ χ ν ′ λ 2 ) 12xλν−24 I ( ν − 2 ) / 2 λxⅇ-12λ+x 1 2 x λ ν 2 4 I ( ν − 2 ) / 2 λ x 1 2 λ x ν+λ ν λ 2ν+2λ 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 m i 1 m i 1 , λ=∑i=1ν m i 2 λ i 1 ν m i 2
Student's t Γν+12νπΓν21+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 Γm+n2Γm2Γn2xm2−11−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 2mnm+n2m+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 Γm+n2Γm2Γn2mnm2xm−221+mnxm+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 2n2m+n−2mn−22n−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 mxmx+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 2nm2m+λ2+m+2λn−2n−22n−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−122NM2 Γ M N2detKK2ⅇ-trK-1w2 w N M 1 2 2 N M 2 Γ M N 2 K K 2 tr K w 2 NK N K covWijWkl=NKikKjl+KilKjk cov W i j W k l N K i k K j l K i l K j k W M NK=∑n=1NxnxnH W M N K n 1 N x n x n , xn∼0K x n 0 K , dimx=M dim x M , Γ M N2=πMM−14∏m=0M−1ΓN2−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 2xaif0≤x≤a21−x1−aifa≤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 λⅇ-λx λ λ x , 0≤x 0 x 1λ 1 λ 1λ2 1 λ 2
Lognormal 12πσ2x2ⅇ-12logx−mσ2 1 2 σ 2 x 2 1 2 x m σ 2 , 0<x 0 x ⅇm+σ22 m σ 2 2 ⅇ2mⅇ2σ2−ⅇσ2 2 m 2 σ 2 σ 2
Maxwell 2πa32x2ⅇ-ax22 2 a 3 2 x 2 a x 2 2 , 0<x 0 x 8πa 8 a 3−8πa-1 3 8 a
Laplacian 12σ2ⅇ-|x−m|σ22 1 2 σ 2 x m σ 2 2 m m σ2 σ 2
Gamma baΓaxa−1ⅇ-bx 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 2axⅇ-ax2 2 a x a x 2 , 0≤x 0 x π4a 4 a 1a1−π4 1 a 1 4
Weibull abxb−1ⅇ-axb a b x b 1 a x b , 0<x 0 x , 0<a∧b 0 a b 1a1bΓ1+1b 1 a 1 b Γ 1 1 b a-2bΓ1+2b−Γ21+1b a 2 b Γ 1 2 b Γ 1 1 b 2
Arc-Sine 1πx1−x 1 x 1 x , 0<x<1 0 x 1 12 1 2 18 1 8
Circular Normal ⅇacosx−m2π 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−maa1+ⅇ-x−ma2 x m a a 1 x m a 2 , 0<a 0 a m m a2π23 a 2 2 3
Gumbel ⅇ-x−maaⅇ-ⅇ-x−ma x m a a x m a , 0<a 0 a m+aγ m a γ a2π26 a 2 2 6
Pareto abax1−a a b a x 1 a , 0<a 0 a ; 0<b≤x 0 b x aba−1 a b a 1 , a>1 a 1 ab2a−2a−12 a b 2 a 2 a 1 2 , a>2 a 2

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