Math 180A - Homework 5 ( Due Friday, Oct 28, 2:00 PM) Reading: Sections 4.1 - 47 of the textbook, and lecture notes 10-14. Show FULL JUSTIFICATION for all your answers. Warm Up Exercises (Do not turn in) 1. Problem 4.25 of the textbook. 2. Problem 4.28 of the textbook. 3. Theoretical Exercise 4.7 of the textbook. 4. Theoretical Exercise 4.8 of the textbook. 5. Suppose each round Bob plays cards with his friends, he wins with probability p, independent of other rounds. Let X be the first round that Bob wins. What are the possible values for X? Find the probability mass function of X. Note: a random variable X with this probability mass function is said to have a Geometric distribution with parameter p. Homework Problems (To turn in) 1. Problem 4.57 of the textbook. 2. A certain typing agency employs 2 typists. For each page typed by typist 1, the probability that there exists an error in that page is .10, independent of other pages. For each page typed by typist 2, the probability that there exists an error on that page is .05, independent of other pages. Your 5-page article is equally likely to be typed by either typist. What is the probability that the article will be typed error free. 3. Problem 4.27 of the textbook. 4. Suppose a random variable X has a Geometric distribution with parameter p (i.e. X Geometric(p), see warm up exercise 5). (a) For every ` 0, compute P (X > `). (b) Find E[X]. 5. Suppose each round Bob plays cards with his friends, he wins with probability p, independent of other rounds. Fix r 0, and let X be the round in which Bob achieves his rth victory. (a) What values can X take? Find the probability mass function of X. (b) Find E[X]. 1 (c) Suppose Boston Celtics beats LA Lakers with probability .6 in each game, independent of their other games. Suppose they play in NBA playoffs, and the first team that wins four games wins the series. Use part (a) to find the probability that Boston Celtics wins the series in the sixth game. (Remark: compare your result with the result we obtain in class.) 6. Problem 4.21 of the textbook. 7. (a) Suppose X is a nonnegative integer-valued random variable (that is, X takes values from {0, 1, 2, 3, ...}). Then, prove that E[X] = X P (X k). (1) k=1 (Hint: Write P (X k) in terms of P (X = `), ` = 1, 2, 3, ..., and then change the order of summations.) (b) Suppose X Geometric(p). Use the equation (1) of part (a), and part (a) of problem 4 to compute E[X]. Make sure your result agrees with what you got in part (b) of problem 4. 8. Problem 4.35 of the textbook. 9. Problem 4.38 of the textbook. 10. Fill out the Questionnaire for week 5 on TritonEd. Problems for more practice (Do not Turn in) 1. Problems 4.31, 4.33, 4.42, 4.46, 4.49, and 4.66 of the textbook. 2. Theoretical Exercises 4.1 and 4.10 of the textbook. 2