Fall 2015 Stat 100A Quiz #3 Due Wednesday, Nov. 25th at 03:00 PM Student's Name:________________________________ SID:__________________ Question (1) EXERCISE 1 A coin is tossed 3 times independently. One of the variables of interest is the number of tails X. Let Y denote the amount of money won on a side bet in the following manner: If the first tail occurs on the first toss, you win $1. If the first tail occurs on the second toss, you win $2. If the first tail occurs on the third toss, you win $3. If no tails appear you lose $1. Construct the joint probability distribution of X and Y . In other words complete the following table where the entries are the probabilities for each pair of values of the variables X and Y . X 0123 -1 ? ? ? ? Y1???? 2???? 3???? EXERCISE 2 A die is rolled and the number observed X is recorded. Then a coin is tossed number of times equal to the value of X. For example if X = 2 then the coin is tossed twice, etc. Let Y be the number of heads observed. Note: Assume that the die and the coin are fair. a. Construct the joint probability distribution of X and Y. b. Find the conditional expected value of Y given X = 5. c. Find the conditional variance of Y given X = 5. EXERCISE 3 Let X and Y denote the proportion of time, out of the workday, that employees I and II, respectively, actually spend performing their assigned tasks. The joint probability density function of X and Y is as follows: + , 0 1; 0 1 (, ) = { 0, 1 1 a) Find ( < , > ) 2 4 b) Find ( + 1) 1