Math 180A - Homework 6 ( Due Friday, Nov 4, 2:00 PM) Reading: Sections 4.3-4.5 and 5.1-5.2 of the textbook. Show FULL JUSTIFICATION for all your answers. Warm-Up Exercises (Do not turn in) 1. Problems 5.1 to 5.3 of the textbook. 2. Theoretical Exercise 5.1 Homework Problems (To turn in) 1. John expects to receive a donation from one of his two donors, with equal probabilities. The amount of each donation is an integer number in dollars. A donation from donor 1 is a random variable with Poisson distribution with parameter 10, while a donation from donor 2 is a random variable with Poisson distribution with parameter 20. Suppose John receives the donation, and observes that the amount is $15. What is the probability that the donation is from donor 1? 2. Earthquakes are measured using Richter magnitude scale. Suppose when an earthquake occurs, the scale of the earthquake is a random variable that can be any positive number {1, 2, 3, 4, ...}, and the probability that the scale of the earthquake is k (k = 1, 2, 3, ..) is equal to c 0.1k , when c is a constant. (a) Find c. (b) Suppose the damage cost of an earthquake with scale k is 10k dollars. When an earthquake occurs, what is the expected damage cost? (c) Now suppose in real life, the scale of an earthquake can only be 1, 2, 3, ..., 9, 10. Repeat parts a and b. 3. Suppose X Poisson(). (a) Show that for every integer n 1, E[X n+1 ] = E[(X + 1)n ]. (1) (Recall that E[X n+1 ] = E[f (X)] when f (x) = xn+1 , and similarly, E[(X +1)n ] = E[g(X)] when g(x) = (x + 1)n .) (b) Using the equation (1) and our calculations for E[X 2 ] and E[X] in class, compute E[X 3 ]. 4. In each round that Bob plays a game, he win probability 0.4. He can choose to play one of these two games: in the first game, he plays 5 rounds, and receives 20 dollars for each win. In the second game, he plays 10 rounds, and receives 10 dollars for each win. (a) What is the expected value of his win in each game? 1 (b) The reward in which game has more uncertainty (i.e. larger variance)? 5. Problem 4.17 of the textbook. 6. Problem 5.4 of the textbook. 7. Problem 5.5 of the textbook. 8. Suppose the cumulative distribution function of a random variable X is given by if x < 0 0 1 x if 0 x < 1 F (x) = 2 1 if x 1. (a) Compute P (X > 1/3). (b) Compute P (X = 0.5) and P (X = 1). (c) Is X a discrete random variable? Is X a continuous random variable? Why? 9. (Optional) You are handed two envelopes, and you know that one of each contains a positive integer dollar amount and that the two amounts are different. Suppose these values are a and b with a < b, and the values of a and b are not known to you. You select at random one of the two envelopes and, after looking at the amount inside, you may switch envelopes if you wish. A friend claims that the following strategy will increase your probability of ending up with an envelope with the larger amount to be greater than 1/2: toss a fair coin repeatedly (assume the tosses are independent), let X be equal to 1/2 plus the number of tosses required to obtain heads for the first time, and switch if the amount in the envelope you selected is less than the value of X. Is your friend correct? More Practice (Do not turn in) 1. Compute the variance of a random variable X, when (a) X has Geometric distribution with parameter p. (b) X has negative binomial distribution with parameters r and p. 2. Theoretical Exercise 5.5. 3. Theoretical Exercise 5.8. 2