Practice Midterm8 - Solutions to Exam#1 ST 8533 Applied...

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Solutions to Exam #1 - ST 8533: Applied Probability Spring, 2001 February 29, 2001 Directions: Answer all questions precisely and completely. To receive full credit, you must de fi ne all notation, and show a reasonable amount of work. 1. (20 points) Giggleswick is a gift store on Main Street in downtown Starkville. Let N represent the number of gifts purchased at Giggleswick, and suppose N has a Poisson distribution with parameter λ . Further suppose that each item purchased will be gift wrapped with probability p . Let X denote the number of items that are wrapped. Use a conditional probability argument to prove that X has a Poisson distribution with parameter λ p . Solution: The proof is as follows: P { X = x } = X n =0 P { X = x | N = n } P { N = n } = X n = x μ n x p x (1 p ) n x e λ λ n n ! = e λ p x X n = x n ! x !( n x )! (1 p ) n x λ n n ! = e λ p x X n = x [ λ (1 p )] n x ( n x )! λ x x ! = ( λ p ) x x ! e λ h e λ (1 p ) i = ( λ p ) x x ! e λ p 2. (20 points) Joe Bob, a small town boy, goes on vacation, and visits “the big city.” He leaves his hotel and walks around to do a little site-seeing. He’s not paying much attention, and all of the sudden realizes he is lost. He comes to an intersection which will take him in three directions. If he chooses Main Street, he will be back at his hotel in 1 hour. If he choose University Avenue, he will walk for 1.5 hours and end up back in the same spot. If he chooses “Big City Circle” he will walk for 2 hours and end up in the same spot. Unfortunately, there are no street signs or distinguishing landmarks at this three way intersection, so at each time, Joe Bob is equally likely to choose any of the three streets.
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