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MATH 1. Identify whether the following statements are true or false. If the statement is false then correct it. (a.) The number of ways to sample k items from n objects with replacement and no regards to order is denoted Ck,n. (b.) Let X and Y be joint random variables with joint PDF, fpx, yq. Let f1pxq be the marginal of X and f2pyq be the marginal of Y . Then gpy|xq “ fpx,yq f2pyq . (c.) Let n and k be integers with n ě k. then Pk,n “ npn ´ 1qpn ´ 2q. . .pn ´ k ´ 1q. (d.) The probability mass function, ppxq, for an experiment gives the probability for each element x in the sample space. (2.) Identify whether the following statements are true or false: (a.) Let S be any sample space and A be an event in S. Then PpAq “ |A| |S| . (b.) Two events A and B are independent if and only if they are mutually exclusive. (c.) Random variables X and Y are joint random variables if and only if there exists function f with fpx, yq “ fXpxqfY pyq for some functions fXpxq and fY pyq. (d.) The collection of events A, B, and C are independent if and only if PpA X B X Cq “ PpAqPpBqPpCq. 1 (3.) Twelve individuals want to form a committee of four. (a.) How many committees are possible? (b.) The 12 individuals consist of 5 biologists, 4 chemists, and 3 physicists. How many committees consisting of 1 biologist, 1 chemist, and 2 physicists are possible? (c.) In the setting of part (b), if all committees are equally as likely, what is the probability the committee formed will consist of 1 biologist, 1 chemist, and 2 physicists? (4.) Let the random variables X and Y have the joint PDF fpx, yq “ kxy2 for 0 ď x ď 2, x ď y ď 3. (a.) Find the constant k. (b.) Find the joint CDF of X and Y . (5.) You are out of town at a research conference; in your suitcase you have two white shirts, 3 black shirts, 2 pairs of black slacks, and 3 pairs of gray slacks. What is the likelihood you wear an all black outfit? (6.) In many states the license plates consist of a string of seven characters such that the first three are letters and the second four are numbers. Assume each string of characters are equally as likely and repetition is allowed on a single license plate. (a.) Describe the random variable, X, that counts the number of vowels in each license plate. (Do not consider y as a vowel.) (b.) Give the probability mass function, ppxq, for this scenario. (c.) Use it to find the likelihood a license plate will have three vowels. (7.) Let X “ t35, 45, 39, 41, 41, 44, 46, 48, 49, 34, 12, 50, 20, 38, 40u (a.) Insert X into R. (a.) Find the mean of X. (b.) Depict X in a boxplot. (NOTE: This question should be answered entirely using code for R.) 2 (8.) Let X have PMF x 1 2 3 4 ppxq 0.4 0.3 0.1 0.2 (a.) Calculate V arpXq and V arp1{Xq. (b.) In a win-win game, the player will win a monetary prize, but has to decide between the fixed prize of $1000/EpXq and the random price of $1000/X, where the random variable X has the PMF given above. Which choice would you recommend the player make? (9.) Consider the bivariate density function fpx, yq “ 12 7 px 2 ` xyq for 0 ď x ď 1 and 0 ď y ď 1. Find the regression function of X on Y . Use it to find the expected value of X given Y “ 0.5. (10.) Two students, A and B, are both registered for a certain course. Assume that student A attends class 80 percent of the time and student B attends class 60 percent of the time, and the absences of the two students are independent. (a.) What is the probability that at least two of the students will be in class on a given day? (b.) It at least one of the two students is in class on a given day, what is the probability that A is in class that day?

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