Table B: Important Continuous Distributions distribution pdf mean variance mgf Uniform ( 1 b−a if x ∈ [a, b] , 0 otherwise b + a 2 (b − a)2 12 ( etb−eta t(b−a) if t = 0 , 1 if t = 0. Standard normal 1 √ 2π e − 1 2 z2 0 1 et 2 /2 Normal 1 σ √ 2π · e− 1 2 ( x−μ σ )2 μ σ2 eμt+σ2t2/2 Exponential λe−λx 1/λ 1/λ2 λ λ − t Gamma λα Γ(α) x α−1 e −λx α/λ α/λ2 » λ λ − t –α Weibull α βα xα−1e−(x/β) α βΓ(1 + 1 α ) β2 h Γ(1 + 2 α ) − ˆ Γ(1 + 1 α ) ˜2 i Beta Γ(α+β) Γ(α)Γ(β) xα−1(1 − x)β−1 α α + β αβ (α + β)2(α + β + 1) Table C: Distributions Derived from the Normal Distributions distribution definition mean variance Chisq(n) = Gamma(α = n 2 , λ = 1 2 ) X 2 = Pn i=1 Zi 2 where Z iid ∼ Norm(0, 1) n 2n F(m, n) F = U/m V /n where U ∼ Chisq(m), V ∼ Chisq(n), and U and V are independent n n − 2 2n2(m + n − 2) m(n − 2)2(n − 4) if n 4 t(n) t = Z p V /n where Z ∼ Norm(0, 1), V ∼ Chisq(n), and Z and V are independent 0 if n 1 n n − 2 if n 2
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