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

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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, , nx x n f xp px n x  =−   " P OISSON { } ( ) !, x fx e x x −µ " H YPERGEOMETRIC {} { } ( ) , max 0, min , , an integer rNr N n r N x rn x xnx n    =+       G AMMA ( 0 , ) x xe x α−1 − β ++ α =∈ = Γαβ \ with 1 0 x x ed x α− Γ(α) ≡ B ETA ( 1 ) , ( 0 , 1 ) ()() x x x υ−1 ω−1 Γ υ+ω ΓυΓω equivalently, (1 ) (, ) xx B υ−1 ω−1 = υω with 1 0 ( 1 ) B d x υ−1 ω−1 υω ≡ E XPONENTIAL ( ) () 1 , [ 0 x e x + ∈∞ = \ N ORMAL 2 2 1 e x p , 2 2 x x x  −− µ  =∀  σ πσ  \
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ARE 210 pdf’s and mgf’s page 2 B IVARIATE N ORMAL 22 2 2 () 2 1 (, ) e x p 2( 1 ) 1 ) yx x y xy xx y y fxy  − σ −µ − ρσ σ −µ +σ  =  σσ −ρ π− ρσσ  2 ∀∈ \ C HI -S QUARED 2 , (2 ) 2 vx v xe fx x v ++ =∈ Γ \ Z i i.i.d. n(0,1) 1 ~( ) v i i Z v = χ . A χ 2 ( v ) is a G( α , β ) with α = v /2 and β = ½.
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This note was uploaded on 08/01/2008 for the course ARE 210 taught by Professor Lafrance during the Fall '07 term at University of California, Berkeley.

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

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