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final - Final Exam Fall 2008 Introduction to Probability...

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Fall 2008 Final Exam Page: 1 of 3 Introduction to Probability Monday, December 16, 2008 STA 4321/5325 Instructions: Please turn of your cell phones. Please write all oF your answers on a separate sheet oF paper and make sure you have clearly labeled the problem corresponding to your answer. Absolutely no cheating. Work as quickly and e±ciently as possible so that you can ²nish all oF the problems. Name: Some Equations Poisson If X Pois( λ ), the pmf of X is p ( x )= λ x x ! e - λ Hypergeometric If X hypergeom( N, k, n ), the pmf of X is p ( x ( k x )( N - k n - x ) ( N n ) and E( X nk/N . Negative Binomial If X NegBin( r, p ), the pmf of X is p ( x ± x + r - 1 r - 1 ² p r (1 - p ) x ,x =0 , 1 , 2 , . . . Also, E( X r (1 - p ) p and var( X r (1 - p ) p 2 . Exponential If X exp( θ ), the pdf of X is f ( x ³ 1 θ e - x/θ 0 0 , else Gamma If X gamma( α, β ), the pdf of X is f ( x ³ 1 Γ( α ) β α x α - 1 e - x/β 0 0 , else Also, E( X αβ and var( X αβ 2 . Beta If X beta( α, β ), the pdf of X is f ( x ³ Γ( α + β ) Γ( α )Γ( β ) x α - 1 (1 - x ) β - 1 , 0 <x< 1 0 , else Also, E[ X ]= α α + β and var( X αβ ( α + β ) 2 ( α + β +1) . Multinomial If X multinomial( n ; p 1 , . . . , p k ), then X has the pmf P ( X 1 = x 1 , . . . , X k = x k n !
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