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MATH 3770 Equations

# MATH 3770 Equations - Poisson Distribution p x E X Pk t e x...

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Poisson Distribution ( 29 ( 29 ( 29 ( 29 ( 29 t k t e t P X V X E x x e x p k t k x α λ = = = = = = - - ! ,... 2 , 1 , 0 , ! ; Expected number of pulses during time interval = αt Probability Density Function (pdf) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 b X a P b X a P b X a P b X a P dx x f b X a P b a < < = < = < = = Cumulative Distributive Function (cdf) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - = = < - = - = = = p dy y f p F p b a a F b F b X a P a F a X P dy y f x X P x F η , 1 If X is a continuous rv with pdf f(x) and cdf F(x) , then at every x at which the derivative F’(x) exists, F’(x) = f(x) ( 29 ( 29 ( 29 ( 29 [ ] ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] ( 29 ( 29 ( 29 ( 29 [ ] 2 2 2 2 2 ~ 5 . X E X E X V X V X E dx x f x X V dx x f x h x h E dx x f x X E F X X X h s - = = - = - = = = = = = = - - - σ μ Normal Distribution ( 29 ( 29 ( 29 - Φ - = - Φ = - Φ - - Φ = - - = - = b b X P a a X P a b b Z a P b Z a P X Z 1 Let X be a binomial rv based on n trials with success probability p . ( 29 ( 29 10 10 5 . , ; - + Φ = = = = nq np npq np x p n x B x X P npq np Gamma Distribution ( 29 - - = Γ 0 1 dx e x x 1. For any 1 , ( 29 ( 29 ( 29 1 1 - Γ - = Γ 2. For any positive integer n , ( 29 ( 29 ! 1 - = Γ n n 3. π = Γ 2 1 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 = = = = = = Γ = - - β αβ , , ; 0 , 0 , 0 , 1 , ; 2 2 1 x F x F x X P X V X E x e x x f x Exponential Distribution ( 29 ( 29 ( 29 ( 29 0 , 1 ; 1 1 0 , 0 , ; 2 2 - = = = = = = - - x e x F X V X E x e x f x x Chi-Squared Distribution Gamma density with 2 ν = and 2 = ( 29 ( 29 ( 29 0 , 2 2 1 ; 2 1 2 2 Γ = - - x e x x f x Weibull Distribution ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 , 1 , ; 1 1 2 1 1 1 0 , 0 , 0 , , ; 2 2 1 - = + Γ - + Γ = + Γ = = - - - x e x F X V X E x e x x f x x Lognormal Distribution ( 29 ( 29 [ ] ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] ( 29 ( 29 - Φ = - = = = - = = = + + - - x x Z P x X P x X P x F e e X V e X E x e x x f x ln ln ln ln , ; 1 0 , 2 1 , ; 2 2 2 2 2 2 2 2 ln Beta Distribution ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 tan 1 0 , 1 , , , ; 2 2 1 1 + + + - = + - + = = = - - - - Γ Γ + Γ - = - A B X V A B A X E dard s B A B x A A B x B A B A x A B B A x f Joint probability mass function: ( 29 [ ] ( 29 ( 29 ( 29 [ ] ( 29 ∫ ∫ ∑∑ = = A y x A dxdy y x f A Y X P y x p A Y X P , , , , , Marginal probability mass functions: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - = = = = dx y x f y f dy y x f x f y x p y p y x p x p Y X x Y y X , , , , Independent if: ( 29 ( 29 ( 29 ( 29

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MATH 3770 Equations - Poisson Distribution p x E X Pk t e x...

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