2.pdf - STATISTICS 244 Problem Set 2 9:00-10:20 TTh Due...

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STATISTICS 244 Problem Set 2 9:00-10:20 TTh Due Thursday January 19 (1) Here are three possible probability density functions for a continuous random variable X; (i) f(x) = 30x 4 (1–x) for 0 x 1 ( = 0 otherwise) (ii) f(x) = e –(x – 0.2) for x 0.2 (= 0 for x < 0.2) (iii) f(x) = ½ sin(x) for 0 x p (= 0 otherwise) In each case, (a) find the cdf of Z; (b) find Pr{0.2 X 0.5}; (c) find E(X); (d) find the median of the distribution of Z; (e) find the probability density of Y = (X + 2) 1/2 . (2) This problem concerns a European style roulette wheel, meaning a wheel with 37 slots numbered 0 to 36. The slots with odd numbers are colored Red, the positive even numbered slots are colored Black, and the Zero slot is colored Green. The order of the numbers (clockwise) is: 0-32-15-19-4-21-2-25-17-34-6-27-13-36-11-30-8-23- 10-5-24-16-33-1-20-14-31-9-22-18-29-7-28-12-35-3-26. The wheel is spun and a single ball is allowed to drop onto the wheel, and is supposed to be equally likely to land in any of the 37 slots, with each spin being independent of the others. (a) What is the probability the ball lands in a Red slot numbered 20 or larger? (b) What is the probability the ball lands in a slot that with a higher number than either of its neighbors? (c) If you win $1 when a Red slot occurs, $2 when a Black slot occurs, and $0 when a Green slot occurs, what is the fair price to play the game?
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