Probability Distributions 1.3 2003/06/03 2003/09/05 15:05:50.926 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Eileen Krause erkrause@rice.edu Kyle Clarkson kclarks@rice.edu Elizabeth Gregory lizzardg@rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Jeanes mjeanes@rice.edu Jeffrey Silverman jsilv@rice.edu Discrete Probability Distributions Name Probability Mean Variance Relationships Discrete Uniform 1 N M 1 M n N 0 M N 2 N M 2 N M 12 Bernoulli n 0 1 p , n 1 p p p 1 p Binomial N n p n 1 p N n , n 0 … N N p N p 1 p Sum of N IID Bernoulli Geometric 1 p p n , n 0 p 1 p p 1 p 2 Negative Binomial n 1 N 1 p N 1 p n N , n N N p N 1 p p 2 Poisson λ n λ n , n 0 λ λ Hypergeometric a n b N n a b N , n 0 … N ; 0 n a b ; 0 N a b N a a b N a b a b N a b 2 a b 1 Logarithmic p n n 1 p p 1 p 1 p p p 1 p 1 p q Distributions Related to the Gaussian Name Probability Mean Variance Relationships Gaussian (Normal) 1 2 σ 2 1 2 x m σ 2 m σ 2 Bivariate Gaussian 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 x m x , y m y x σ x 2 , y σ y 2 , x y m x m y ρ σ x ρ σ y ρ: correlation coefficient Conditional Gaussian 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 σ x 2 1 ρ 2 Generalized Gaussian 1 2 Γ 1 1 r A r x m A r r m σ 2 A r σ 2 Γ 1 r Γ 3 r Chi-Squared ( χ ν 2 ) 1 2 ν 2 Γ ν 2 x ν 2 1 x 2 , 0 x ν 2 ν χ ν 2 i 1 ν x i 2 , x i IID 0 1 Noncentral Chi-Squared ( χ ν ′ λ 2 ) 1 2 x λ ν 2 4 I ( ν − 2 ) / 2 λ x 1 2 λ x ν λ 2 ν 2 λ χ ν ′ 2 i 1 ν x i 2 , x i IID m i 1 , λ i 1 ν m i 2 Student's t Γ ν 1 2 ν Γ ν 2 1 x 2 ν ν 1 2 0 ν ν 2 , 2 ν Beta β m , n Γ m n 2 Γ m 2 Γ n 2 x m 2 1 1 x n 2 1 , 0 x 1 , 0 m n m m n 2 m n m n 2 m n 2 β m , n χ m 2 χ m 2 χ n 2 F Distribution 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 , 1 m n n n 2 , n 2 2 n 2 m n 2 m n 2 2 n 4 , n 4 F m , n χ m 2 m χ n 2 n Non-central F [ F m , n ′ λ ] k 0 λ 2 k k λ 2 p β m 2 + k , n 2 m x m x n n n 2 , n 2 2 n m 2 m λ 2 m 2 λ n 2 n 2 2 n 4 , n 4 F m , n ′ λ χ m ′ λ 2 m χ n λ 2 n Wishart W M N K w N M 1 2 2 N M 2 Γ M N 2 K K 2 tr K w 2 N K cov W i j W k l N K i k K j l K i l K j k W M N K n 1 N x n x n , x n 0 K , dim x M , Γ M N 2 π M M 1 4 m 0 M 1 Γ N 2 m 2 Non-Gaussian Distributions Name Probability Mean Variance Relationships Uniform 1 b a , a x b a b 2 b a 2 12 Triangular 2 x a 0 x a 2 1 x 1 a a x 1 1 a 3 1 a a 2 18 Exponential λ λ x , 0 x 1 λ 1 λ 2 Lognormal 1 2 σ 2 x 2 1 2 x m σ 2 , 0 x m σ 2 2 2 m 2 σ 2 σ 2 Maxwell 2 a 3 2 x 2 a x 2 2 , 0 x 8 a 3 8 a Laplacian 1 2 σ 2 x m σ 2 2 m σ 2 Gamma b a Γ a x a 1 b x , 0 x , 0 a b a b a b 2 Rayleigh 2 a x a x 2 , 0 x 4 a 1 a 1 4 Weibull a b x b 1 a x b , 0 x , 0 a b 1 a 1 b Γ 1 1 b a 2 b Γ 1 2 b Γ 1 1 b 2 Arc-Sine 1 x 1 x , 0 x 1 1 2 1 8 Circular Normal a x m 2 I 0 a , x m Cauchy a x m 2 a 2 m (from symmetry arguments) Logistic x m a a 1 x m a 2 , 0 a m a 2 2 3 Gumbel x m a a x m a , 0 a m a γ a 2 2 6 Pareto a b a x 1 a , 0 a ; 0 b x a b a 1 , a 1 a b 2 a 2 a 1 2 , a 2