practice_midterm2 - STAT 230 Introduction to Probability...

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Unformatted text preview: STAT 230 Introduction to Probability and Random Variables Summer 2009 Practice for Midterm 2 Exercise Let X be the number of fish caught by a fisherman in one afternoon. Suppose that X is distributed Poisson ( λ ) . Each fish has probability p of being a salmon independently of all other fish caught. Let Y be the number of salmon caught. Show that Y is Poisson ( p λ ) . ( ) ( | ) ( ) (1 ) ! (1 ) ! ! (1 ) !( )! ! (1 ) 1 (1 ) ! ( )! (1 ) x y x y x y x y x y x y x y x y x y x y y y x x x y y y P Y y P Y y X x P X x x p p e y x x p p e y x x p p y x y x p p e p y x y p p y λ λ λ λ λ λ λ ∞ = ∞ − − = ∞ − − = ∞ − = − ∞ − = − = = = = = = − × = − × = − × − − = − × − − = ∑ ∑ ∑ ∑ ∑ (1 ) 1 (1 ) ! !...
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practice_midterm2 - STAT 230 Introduction to Probability...

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