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Midterm 2 (Stor 435)
November 8, 2007 The midterm has 10 problems. Each problem is worth 3 points. Show all the work and write in the space provided. If more space is needed, write on the back of the pages. Problem 1. Three balls are randomly chosen from an urn containing 3 white, 3 red and 5 black balls. Suppose that we win $1 for each white ball selected and lose $1 for each red selected. (For each black, we neither win nor lose.) Let X denote our total winnings from the experiment. What are the possible values of X , and what are the probabilities associated with each value? Solution: Problem 2. A contestant on a quiz show is presented with two questions, questions 1 and 2, which he is to attempt to answer in some order chosen by him. If he decides to try question i ﬁrst, then he will be allowed to go on to question j , j = i, only if his answer to question i is correct. If his initial answer is not correct, he is not allowed to answer the other question. The contestant is to receive 200 and 100 dollars if he correctly answers questions 1 and 2, respectively. Thus, for instance, he will receive 300 dollars if both questions are correctly answered. If the probabilities that he knows the answers to questions 1 and 2 are 0.6 and 0.8, respectively, which question he should attempt ﬁrst so as to maximize his expected winnings? (Assume that the events Ei , i = 1, 2, that he knows the answer to question i, are independent events.) Solution: 1 Problem 3. Consider a jury trial in which it takes 8 of the 12 jurors to convict; that is, in order for the defendant to be convicted, at least 8 of the jurors must vote him guilty. Assume that jurors act independently and each makes the right decision (that is, if guilty, ﬁnds guilty, and if innocent, ﬁnds innocent) with probability 0.8. If 0.7 represents the probability that the defendant is guilty, what is the probability that the jury renders a correct decision? (You can leave you answer in a numerical expression - no need to provide an actual numerical value.) Solution: Problem 4. People enter a gambling casino at a rate of 1 for every 2 minutes. What is the probability that at least 4 people enter the casino between 12:00 and 12:07? (Use the Poisson probabilities.) Solution: Problem 5. Let Y = B (n, p) be a binomial random variable. If n is large, p is small and λ = np is moderate, argue that the binomial probability P (Y = i) = n pi (1 − p)n−i is approximately the i 1 Poisson probability P (Pois(λ) = i) = e−λ λi /i!. (Recall that (1 − y )y ≈ e−y for large y .) Solution: 2 Problem 6. Let X be a continuous random variable with a cumulative distribution function (c.d.f.) 0, x ≤ 0, 2 − 2x3 , 0 < x < 1, 3x F (x) = 1, x ≥ 1. (a) Find the density function f of X . (b) Express the probability P (|X − 1/2| > 1/3) in terms of the c.d.f. F . Solution: Problem 7. For the random variable X in Problem 6 above, ﬁnd and draw the c.d.f. FY and the √ density function fY of the random variable Y = 2 X . Solution: Problem 8. The density function of X is given by f (x) =
a + bx2 , 0 < x < 1, 0, otherwise. (a) If EX = 3/5, ﬁnd a and b. (b) Compute P (X > 1/2). Solution: 3 Problem 9. A model for the movement of a stock supposes that if the present price of the stock is s, then after one time period it will be either u · s with probability p, or d · s with probability 1 − p. Assuming that successive movements are independent, use normal distribution to approximate the probability that the stock price will be up at least 30 percent after the next 1,000 time periods if u = 1.012, d = 0.990, and p = .52. Solution: Problem 10. A ﬁre station is to be located along a road of inﬁnite length - stretching from point 0 outward to ∞. If the distance of a ﬁre from point 0 is exponentially distributed with parameter λ = 1, where should the ﬁre station be located so as to minimize the expected distance from the ﬁre? That is, choose a as to minimize E |X − a| when X is exponential with parameter λ = 1. Solution: 4 ...
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This note was uploaded on 05/11/2011 for the course STOR 435 taught by Professor Shakar during the Spring '11 term at University of North Carolina School of the Arts.
- Spring '11