Statistics- Fall 2015 Stat 100A Quiz #3

QUESTION

Fall 2015Stat 100AQuiz #3Due Wednesday, Nov. 25th at 03:00 PMStudent’s Name:________________________________SID:__________________Question (1)EXERCISE 1A 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 followingtable where the entries are the probabilities for each pair of values of the variables X and Y .X0123-1 ? ? ? ?Y1????2????3????EXERCISE 2A die is rolled and the number observed X is recorded. Then a coin is tossed number of timesequal to the value of X. For example if X = 2 then the coin is tossed twice, etc. Let Y be thenumber 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 3Let 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 densityfunction of X and Y is as follows:+ ,0 ≤ ≤ 1; 0 ≤ ≤ 1(, ) = {0,ℎ11a) Find ( < , > )24b) Find ( + ≤ 1)1

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