spring98exam2[1]

# Therefore the probability of error is just the

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Unformatted text preview: e probability of error is just the probability that H0 occurs, i.e., P (H0 ) = 0:2. Pe = 0:2 Problem 3 (34 points) Alice is a \professional" dart player, who is invited to play in a tournament. The tournament involves throwing darts at a dartboard, shown in the diagram below. Each contestant throws darts at the dartboard, one dart after another, as long as he or she hits the bullseye, and leaves the tournament after the rst miss of the bullseye. Bullseye 1 1 1 Let X denote the random variable corresponding to the distance from the center that Alice hits on each dart throw. The probability density function of X is given by 8 4u if 0 u < 1:5 >9 < fX (u) = > 4 ; 4 u if 1:5 u < 3 :3 9 0 otherwise The bullseye is the innermost circle of radius 1 about the center of the dartboard. a. Let A be the event that Alice hits the bullseye on a given throw. What is the probability of Answer: P (A) is just the integral of fX (u) over the range 0 u 1. This is just the area of a triangle of height 4=9 and base length 1, and is equal to 2=9. A? P (A) = 2 9 b. Let Y denote the total number of darts Alice gets to throw in the tournament, before she leaves after her rst miss. Assume that successive throws are independent trials. Find the expected value and the variance of Y . eter (p) equal to Alice's probability of missing the bullseye, i.e., p = 7=9. 1 ; 2=9 Therefore E Y ] = p = 9 and Var Y ] = 1 p2 p = 49=81 7 Answer: The key here is to realize that Y is a geometric random variable with param- E Y] = 9 7 Var Y ] = 18 49 c. Alice is paid \$20 for taking part in the tournament and \$70 for each dart she gets in the bullseye. Let Z deno...
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