Table of Common Distributions taken from Statistical Inference by Casella and Berger Discrete Distrbutions distribution pmf mean variance mgf/moment Bernoulli(p) px (1 − p)1−x ; x = 0, 1; p ∈ (0, 1) p p(1 − p) (1 − p) + pet Beta-binomial(n, α, β) Γ(α+β) Γ(x+α)Γ(n−x+β) (nx ) Γ(α)Γ(β) Γ(α+β+n) nα α+β nαβ (α+β)2 Notes: If X|P is binomial (n, P ) and P is beta(α, β), then X is beta-binomial(n, α, β). Binomial(n, p) ( nx )px (1 − p)n−x ; x = 1, . . . , n np np(1 − p) Discrete Uniform(N ) 1 N; N +1 2 (N +1)(N −1) 12 [(1 − p) + pet ]n PN it 1 i=1 e N Geometric(p) p(1 − p)x−1 ; p ∈ (0, 1) 1 p 1−p p2 pet 1−(1−p)et x = 1, . . . , N Note: Y = X − 1 is negative binomial(1, p). The distribution is memoryless: P (X > s|X > t) = P (X > s − t). Hypergeometric(N, M, K) N −M (M x )( K−x ) ; N ( K) x = 1, . . . , K KM N KM (N −M )(N −k) N N (N −1) r(1−p) p r(1−p) p2 λ λ M − (N − K) ≤ x ≤ M ; N, M, K > 0 Negative Binomial(r, p) )pr (1 − p)x ; p ∈ (0, 1) (r+x−1 x ? ³ p 1−(1−p)et r y−r ( y−1 ; Y =X +r r−1 )p (1 − p) Poisson(λ) e−λ λx x! ; λ≥0 Notes: If Y is gamma(α, β), X is Poisson( βx ), and α is an integer, then P (X ≥ α) = P (Y ≤ y). 1 t eλ(e −1) ´r Continuous Distributions distribution pdf Beta(α, β) Γ(α+β) α−1 (1 Γ(α)Γ(β) x Cauchy(θ, σ) 1 1 πσ 1+( x−θ )2 ; σ − x)β−1 ; x ∈ (0, 1), α, β > 0 σ>0 mean variance α α+β αβ (α+β)2 (α+β+1) mgf/moment P∞ ³Qk−1 1 + k=1 r=0 does not exist does not exist does not exist Notes: Special case of Students’s t with 1 degree of freedom. Also, if X, Y are iid N (0, 1), χ2p Notes: Gamma( p2 , 2). Double Exponential(µ, σ) Exponential(θ) p x 2 −1 e− 2 ; x > 0, p ∈ N 1 p 2 Γ( p 2 )2 x 1 − |x−µ| σ ; 2σ e 1 −x θ ; x ≥ θe σ>0 0, θ > 0 q 1 p 2p µ 2σ 2 θ θ2 X Y is Cauchy ³ 1 1−2t ´ p2 eµt 1−(σt)2 1 1−θt , t X Notes: Gamma(1, θ). Memoryless. Y = X γ is Weibull. Y = 2X β is Rayleigh. Y = α − γ log β is Gumbel. ³ ´ ν21 ν1 −2 ν +ν Γ( 1 2 2 ) ν1 ν2 +ν2 −2 x 2 2 ; x>0 2( ν2ν−2 )2 νν11 (ν , ν2 > 4 Fν1 ,ν2 ν1 ν2 ¡ ν1 ¢ ν1 +ν 2 ν2 −2 , ν2 > 2 Γ( 2 )Γ( 2 ) ν2 2 −4) 2 , t< Γ( 1+( ν )x ´ tk k! 1 2 1 θ < EX n = α+r α+β+r ν1 +2n ν −2n )Γ( 2 2 ) 2 ν ν Γ( 21 )Γ( 22 ) ³ ν2 ν1 ´n , n< 2 Notes: Fν1 ,ν2 = χ2ν1 /ν1 χ2ν2 /ν2 , 2 where the χ s are independent. F1,ν = t2ν . x 1 α−1 − β e ; Γ(α)β α x αβ 2 q Notes: Some special cases are exponential (α = 1) and χ2 (α = p2 , β = 2). If α = 23 , Y = X β is Maxwell. Y = Gamma(α, β) Logistic(µ, β) 1 β h − e x−µ β − 1+e x > 0, α, β > 0 i2 ; β > 0 x−µ β 1 Notes: The cdf is F (x|µ, β) = − 1+e x−µ β ³ αβ 1 X π2 β 2 3 µ 1 1−βt ´α , t< 1 β is inverted gamma. eµt Γ(1 + βt), |t| < 1 β . (log x−µ)2 2σ 2 σ2 2 n2 σ 2 2 Lognormal(µ, σ 2 ) 1 − √1 e 2πσ x Normal(µ, σ 2 ) √ 1 e− 2πσ Pareto(α, β) βαβ ; xβ+1 x > α, α, β > 0 βα β−1 , tν Γ( ν+1 2 ) Γ( ν2 ) √1 νπ 0, ν > 1 ν ν−2 , ν b+a 2 (b−a)2 12 ebt −eat (b−a)t h i 2 β γ Γ(1 + γ2 ) − Γ2 (1 + γ1 ) EX n = β γ Γ(1 + nγ ) Notes: t2ν = F1,ν . (x−µ)2 2σ 2 ; x > 0, σ > 0 ; σ>0 1 2 (1+ xν ν+1 ) 2 1 Uniform(a, b) b−a , a ≤ x ≤ b Notes: If a = 0, b = 1, this is special case of beta (α = β = 1). xγ γ γ−1 − β e ; x > 0, γ, β > 0 Weibull(γ, β) βx Notes: The mgf only exists for γ ≥ 1. eµ+ 2 e2(µ+σ ) − e2µ+σ 2 σ2 µ 1 β>1 β γ Γ(1 + γ1 ) 2 EX n = enµ+ eµt+ βα2 (β−1)2 (β−2) , β>2 >2 σ 2 t2 2 does not exist EX n = n n Γ( ν+1 2 )Γ(ν− 2 ) √ ν2, πΓ( ν2 ) n n even ν2 2
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