Math 180A - Homework 9 ( Due Monday Nov 28, 2:00 PM) Reading: Sections 6.1 - 6.3, 7.2 and 7.4 of the textbook. Show FULL JUSTIFICATION for all your answers. 1 Warm up Problems - Do not turn in 1. Problems 6.2, 6.8, 6.10, 6.11 of the textbook. 2. Problems 7.1, 7.4, and 7.6 of the textbook. 2 Homework Problems - To turn in 1. Problem 6.9 of the textbook. 2. Problem 6.7 of the textbook. 3. Two fair dice are rolled. Let X be the value of the first die, and Y be the value of the second die. (a) Find the joint probability mass function of X and Y . Are X and Y independent? (b) Let S be the smallest and L be the largest value obtained on the dice. find the joint probability mass function for S and L. (c) Find the (marginal) probability mass functions for S and L. Are S and L independent? 4. (This is the example we looked at briefly in class.) Suppose X and Y are continuous random variables with probability density function ( 1 2 xy if 0 < x < y < 2, fX,Y (x) = 0 otherwise . (a) Verify that fX,Y is a joint probability density function. (b) Compute the marginal p.d.f.s of X and Y . Are X and Y independent? For the next problem, recall that a collection of random variables X1 ,...,Xn are called independent, if for all subsets I1 , ..., In of R, the events {X1 I1 }, ..., {Xn In } are independent. 5. Suppose U1 , U2 , ..., Un are independent random variables and for every i = 1, ..., n, Ui has a uniform distribution over [0, 1]. (a) Find the probability density function of M = max(U1 , ..., Un ). (We solved the case n = 2 in class. This a generalization.) (b) Define Z = min(U1 , ..., Un ). Find the c.d.f. and the p.d.f of Z. (Hint: write the event {Z > z} in terms of random variables U1 , ..., Un .) 6. Problem 6.29 of the textbook. 1 7. Problem 7.5 of the textbook. 8. Suppose a car dealer earns $1000 on each used car sold, and earns $2000 on each new car sold. In a month, the number of used cars sold has a mean of 10 with a standard deviation of 5, and the number of new cars sold has a mean of 20, with a standard deviation of 8. Suppose the number of used cars sold, and the number of new cars sold are independent. Find the mean and standard deviation of the dealer's profit in a month. 9. Suppose X has an exponential distribution with parameter , and Y has an exponential distribution with parameter . Suppose X and Y are independent. (a) Find P (X > Y ). (b) Compute E[ X Y ] (c) Find the p.d.f. of X + Y . 3 More Practice - Do not turn in 1. Use induction to prove that for any number of random variables X1 ,...,Xn , E[X1 + ... + Xn ] = E[X1 ] + ... + E[Xn ]. (Note: We proved the base case n = 2 in class. ) 2. Problem 6.19 of the textbook. 3. Theoretical Exercise 6.6 (hint: find the c.d.f first), 6.9 and 6.11 of the textbook. 4. Prove that if X N (1 , 12 ) and Y N (2 , 22 ), then X + Y has a normal distribution with mean 1 + 2 and variance 12 + 22 . 2 fX,Y (x, y) = 1 36 2 36 1 fS,L (s, `) = 36 0 if s < ` if s = ` if s > ` 1 fX,Y (x, y) = 1 36 2 36 1 fS,L (s, `) = 36 0 if s < ` if s = ` if s > ` 1