pdf-formulas - ARE 210 pdf's and mgf's page 1 PROBABILITY...

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ARE 210 pdf’s and mgf’s page 1 P ROBABILITY D ENSITY F UNCTIONS B INOMIAL ( ) { } ( ) 1 , 0,1,2, , n x x n f x p p x n x = " P OISSON { } ( ) !, 0,1,2, x f x e x x −µ = µ " H YPERGEOMETRIC { } { } ( ) , max 0, min , , an integer r N r N f x n r N x r n x x n x n  = +   G AMMA ( ) (0, ) ( ) x x e f x x α−1 β ++ α = ∞ = Γ α β \ with 1 0 x x e dx α− Γ(α) ≡ B ETA ( ) ( ) (1 ) , (0,1) ( ) ( ) f x x x x υ−1 ω−1 Γ υ+ ω = Γ υ Γ ω equivalently, (1 ) ( ) ( , ) x x f x B υ−1 ω−1 = υ ω with 1 0 ( , ) (1 ) B x x dx υ−1 ω−1 υ ω ≡ E XPONENTIAL ( ) ( ) 1 , [0, ) x f x e x µ + = µ ∞ = \ N ORMAL ( ) 2 2 1 ( ) exp , 2 2 x x f x x −µ = σ πσ \
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ARE 210 pdf’s and mgf’s page 2 B IVARIATE N ORMAL 2 2 2 2 2 2 2 2 ( ) 2 ( )( ) ( ) 1 ( , ) exp 2 (1 ) 2 (1 ) y x x y x y x y x y x y x x y y f x y − σ −µ ρσ σ −µ −µ + σ −µ = σ σ −ρ π −ρ σ σ 2 ( , ) x y
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