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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 {1NM+1  if  MnN0  otherwise   1 N M 1 M n N 0 M+N2 M N 2 (NM+2)(NM)12 N M 2 N M 12
Bernoulli Prn=0=1p n 0 1 p , Prn=1=p n 1 p p p p(1p) p 1 p
Binomial Nnpn1pNn N n p n 1 p N n , n=0N n 0 N Np N p Np(1p) N p 1 p Sum of NN IID Bernoulli
Geometric (1p)pn 1 p p n , n0 n 0 p1p p 1 p p1p2 p 1 p 2
Negative Binomial n1N1pN1pnN n 1 N 1 p N 1 p n N , nN n N Np N p N(1p)p2 N 1 p p 2
Poisson λneλn! λ n λ n , n0 n 0 λ λ λ λ
Hypergeometric anbNna+bN a n b N n a b N , n0N n 0 N ; 0na+b 0 n a b ; 0Na+b 0 N a b Naa+b N a a b Nab(a+bN)a+b2(a+b1) N a b a b N a b 2 a b 1
Logarithmic pnnlog(1p) p n n 1 p p(1p)log(1p) p 1 p 1 p (p)(p+log(1p))(1p)logq p p 1 p 1 p q
Table 2: Distributions Related to the Gaussian
Name Probability Mean Variance Relationships
Gaussian (Normal) 12πσ2e(12xmσ2) 1 2 σ 2 1 2 x m σ 2 m m σ2 σ 2
Bivariate Gaussian 12π1ρ2 σ x σ y e(12×(1ρ2)(x m x σ x 22ρx 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 , 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 =12π1ρ2 σ x 2ex m x ρ σ x σ y (y m y )22 σ x 2×(1ρ2) 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(1ρ2) σ x 2 1 ρ 2
Generalized Gaussian 12Γ1+1rAre|xmAr|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 ) 12ν2Γν2xν21ex2 1 2 ν 2 Γ ν 2 x ν 2 1 x 2 , 0x 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 𝒩01 0 1
Noncentral Chi-Squared ( χ ν 2λ χ ν λ 2 ) 12xλν24 I ( ν 2 ) / 2 (λx)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+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 𝒩 m i 1 m i 1 , λ= i =1ν m i 2 λ i 1 ν m i 2
Student's t Γν+12νπΓν21+x2νν+12 Γ ν 1 2 ν Γ ν 2 1 x 2 ν ν 1 2 0 0 νν2 ν ν 2 , 2<ν 2 ν
Beta β m , n β m , n Γm+n2Γm2Γn2xm211xn21 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<mn 0 m n mm+n m m n 2mnm+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 Γm+n2Γm2Γn2mnm2xm221+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 , 0x 0 x , 1mn 1 m n nn2 n n 2 , n>2 n 2 2n2(m+n2)mn22(n4) 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 =0λ2kk!eλ2 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 nn2 n n 2 , n>2 n 2 2nm2m+λ2+(m+2λ)(n2)n22(n4) 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 detwNM122NM2 Γ M N2detKK2etrK-1w2 w N M 1 2 2 N M 2 Γ M N 2 K K 2 tr K w 2 NK N K covWi,jWk,l=N(Ki,kKj,l+Ki,lKj,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 , dimx=M dim x M , Γ M N2=πM(M1)4 m =0M1ΓN2m2 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 1ba 1 b a , axb a x b a+b2 a b 2 ba212 b a 2 12
Triangular {2xa  if  0xa2×(1x)1a  if  ax1 2 x a 0 x a 2 1 x 1 a a x 1 1+a3 1 a 3 1a+a218 1 a a 2 18
Exponential λe(λx) λ λ x , 0x 0 x 1λ 1 λ 1λ2 1 λ 2
Lognormal 12πσ2x2e(12logxmσ2) 1 2 σ 2 x 2 1 2 x m σ 2 , 0<x 0 x em+σ22 m σ 2 2 e2m(e2σ2eσ2) 2 m 2 σ 2 σ 2
Maxwell 2πa32x2eax22 2 a 3 2 x 2 a x 2 2 , 0<x 0 x 8πa 8 a (38π)a-1 3 8 a
Laplacian 12σ2e|xm|σ22 1 2 σ 2 x m σ 2 2 m m σ2 σ 2
Gamma baΓaxa1e(bx) b a Γ a x a 1 b x , 0<x 0 x , 0<ab 0 a b ab a b ab2 a b 2
Rayleigh 2axe(ax2) 2 a x a x 2 , 0x 0 x π4a 4 a 1a(1π4) 1 a 1 4
Weibull abxb1e(axb) a b x b 1 a x b , 0<x 0 x , 0<ab 0 a b 1a1bΓ1+1b 1 a 1 b Γ 1 1 b a2b(Γ1+2bΓ21+1b) a 2 b Γ 1 2 b Γ 1 1 b 2
Arc-Sine 1πx(1x) 1 x 1 x , 0<x<1 0 x 1 12 1 2 18 1 8
Circular Normal eacosxm2π I 0 a a x m 2 I 0 a , π<xπ x m m
Cauchy aπxm2+a2 a x m 2 a 2 m m (from symmetry arguments)
Logistic exmaa1+exma2 x m a a 1 x m a 2 , 0<a 0 a m m a2π23 a 2 2 3
Gumbel exmaaeexma x m a a x m a , 0<a 0 a m+aγ m a γ a2π26 a 2 2 6
Pareto abax1a a b a x 1 a , 0<a 0 a ; 0<bx 0 b x aba1 a b a 1 , a>1 a 1 ab2(a2)a12 a b 2 a 2 a 1 2 , a>2 a 2

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