# Each week, the number of hours Oscar spends working is uniformly distributed between zero and one hundred. He gets paid one dollar an hour. If Oscar…

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Q13. Each week, the number of hours Oscar spends working is uniformly distributedbetween zero and one hundred. He gets paid one dollar an hour. If Oscar works more thanﬁfty hours in a week, there is a probability of 1 / 2 he will be paid overtime, receiving twodollars an hour for each hour beyond the initial ﬁfty. Independently of receiving overtimepay, if Oscar works more than seventy ﬁve hours in a week, there is a probability of 1 / 2that he will receive a one hundred dollar bonus. Find the expected value and variance ofOscar’s weekly pay. Q14. Let X be a geometric random variable with parameter P, where P is itself random anduniformly distributed from 0 to (n — 1) / n. Let Z = lE[X|P]. Find lE[Z] and limn_,00 lE[Z]. Q15. The random variables X and Y are described by a joint PDF which is constant withinthe unit area quadrilateral with vertices (0,0), (0, 1), (1,2), and (1,1). Use the law of totalvariance to ﬁnd the variance of X + Y.