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Summary+of+Probability+Distributions

Summary+of+Probability+Distributions - Name of Distribution...

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Summary of Probability Distributions of Discrete Random Variables Name and parameter(s) of the distribution pmf( * ) Mean X ) Variance (σ 2 ) mgf Binomial B(n,p) 1 x n x n p( x ) p ( p ) x -   = -     x = 0, 1, …, n np np(1-p) [pe t + (1-p)] n Bernoulli n =1, p Negative Binomial nb(r, p) 1 1 r x r x p( x ) p ( p ) x - - = - x = r, r + 1, … r/p r(1-p)/p 2 1 1 r t r ( p )e - - Geometric r = 1, p Hypergeometric X ~ h(N, M, n) M N M N p( x ) / x n x n -    =    -    x = 0, 1, …, k and M – (N – k) ≤ x ≤ M M n( ) N 1 1 M M N n n( )( )( ) N N N - - - Not derived Poisson X ~ Π (λ) x p( x ) e / x! λ - = x = 0, 1, 2, … λ λ 1 t ( e ) e - ( * ) Note that p(x) = 0 outside the ranges specified under each pmf.
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Summary of Probability Distributions of Continuous Random Variables
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Unformatted text preview: Name of Distribution pdf( * ) Mean (μ X ) Variance (σ 2 ) mgf Beta Βeta(α, β ) 1 1 1 ( ) f ( x ) x ( x ) ( ) ( ) α β--Γ + =-Γ Γ 0 ≤ x ≤ 1 + 2 1 ( ) ( ) αβ + + + Too complicated!!! U(0,1) α = β = 1 Gamma G(α, β ) 1 1 x / f ( x ) x e ( a )--= Γ x > 0 α β α β 2 (1 – β t)- α Exp( θ ) α=1, β = θ . 2 χ ( n ) α=n/2, β = 2 Normal X ~ N(μ, σ 2 ) 2 2 2 1 2 ( x ) f ( x ) e μ σ π σ-= μ σ 2 2 2 1 2 t t e + ( * ) Note that F(x) = 0 outside the ranges specified under each pdf....
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