STAT / MATH 414 Final (Total points: 45) The following rules apply. 1. It is openbook and you can use anything (including books, calculator, and internet). 2. You may not collaborate or copy. Failure to comply with this rule could lead to reduction in your grade, or disciplinary action. By taking this nal exam, you agree that you have read the rules above and agree to comply with them. Q1 [4pts] A salesman makes random calls to sell a product. A call is said to be \"good\" if the salesm Y dened [(A), (B), a. 1pts b 1pts the e. 1pts d. 1pts Q2 [1 1pts] Th 3. [2pts] b. [3pts] c. [3pts] d. [3pts] an succeeds in selling his / her product on this call. For the random variable in each of the following 4 cases, please select one type of random variables (C), (D)7 (E), (F)] from the options below that best describes Y. (Your answer: ) The salesman makes 100 calls and Y is the number of \"good\" calls. (Your answer: ) The salesman continuously makes call until rst \"good\" call. Y is the total number of calls. (Your answer: ) The salesman continuously makes call until he/ she gets 10 \"good\" calls. Y is the total number of calls. (Your answer: ) Y is the total number of typos in a book. e density of Y is given by try 1 i y , 0 S y S 1 f (y) = ( ) 0, elsewhere Find 16 that makes f (y) a valid density function. Find P(Y g 0.5). Find P(Y 2 0.3|Y g 0.5) Find the variance Var(5 * 3Y). Q3 [14pts] Suppose that Y1 and Y2 have joint density function given by 691, 0y23y1 2). (b) [3pts] Find the marginal density f1(y1) of Y]. (c) [3pts] Find the conditional probability density function f(y2l'y1) of Y2 given Y1 = y1- (d) [3pts] Find P(Y2 g 1|Y1 : 2). (e) [Zpts] Are Y1 and Y2 independent? Q4 [6pts] A11 unfair coin has a probability of p E (0, 1) landing tail and a probability of l i p landing head. Repeatedly ip the coin until we observe the pattern HTHH. Let X be the total number of ips needed. Find the expectation E(X). It is enough to write down the equations without solving them. Q5 [4pts] Let Y be a random variable with density function f(y):{3y2, 032/31 0. elsewhere a. [2pts] Find the probability density function of U = 3 2Y. b. [2pts] Find the probability density function of V : Y2. Q6 [Gpts] The downtime per day for a computing facility has mean ,u : 4 hours and standard deviation 0 : 2 hours. a. [3pts] Find the approximate probability that the average daily downtime for a period of 100 days is between 3.6 and 4.3 hours. b. [3pts] What is the (approximate) probability that the TOTAL downtime for a period of 100 days exceeds 420 hours