9assignment - Compute the expected number of matches that...

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ECEN 303: Assignment 9 Problems: 1. The county hospital is located at the center of a square whose sides are 3 miles wide. If an accident occurs within this square, then the hospital sends out an ambulance. The road network is rectangular, so the travel distance from the hospital, whose coordinates are (0 , 0), to the point ( x, y ) is | x | + | y | . If an accident occurs at a point that is uniformly distributed in the square, Fnd the expected travel distance of the ambulance. 2. Let Z be a standard normal random variable, and for a Fxed x , set X = b Z if Z > x 0 otherwise . Show that E[ X ] = 1 2 π e - x 2 / 2 . 3. Cards from an ordinary deck of 52 playing cards are turned face up one at a time. If the Frst card is an ace, or the second a deuce, or the third a three, or, . . . , or the thirteenth a king, or the fourteenth an ace, and so on, we say that a match occurs. Note that we do not require that (13 n + 1)th card be any particular ace for a match to occur but only that it be an ace.
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Unformatted text preview: Compute the expected number of matches that occur. 4. If X 1 , X 2 , . . ., X n are independent and identically distributed random variables having uni-form distributions over (0 , 1), Fnd (a) E[max( X 1 , . . ., X n )]; (b) E[min( X 1 , . . ., X n )]. 5. If X and Y are independent and identically distributed with mean and variance 2 , Fnd E B ( X-Y ) 2 . 6. Ten hunters are waiting for ducks to y by. When a ock of ducks ies overhead, the hunters Fre at the same time, but each chooses his target at random, independently of the others. If each hunter independently hits his target with probability 0.6, compute the expected number of ducks that are hit. Assume that the number of ducks in a ock is a Poisson random variable with mean 6. 1...
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