Hey i need help with problem 3,5 and 6. Please if you can show work, I will appreciate it.

Annie and Alvie have agreed to meet between 5:00 PM. and 6:00 P.M. for dinner at a local health-food restaurant. Let X = Annie's arrival time and Y = Alvie's arrival time. Suppose X and Y are independent with each uniformly distributed on the interval [5, 6]. (a) What is the joint pdf ofX and Y? ' 55x56,55y$6 KXJ') = ' othenNise (b) What is the probability that they both arrive between 5:15 and 5:45? (c) If the rst one to arrive will wait only 30 min before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? [Hint: The event of Interest is A = {(x, y): |x y| 5 1/2}. ] (Round your answer to three decimal places.) Need Help? ii A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf ofX and Y appears in the accompanying tabulation. Y p(x, y) 0 1 2 0 0.10 0.03 0.01 x 1 0.06 0.20 0.07 2 0.05 0.14 0.34 (a) Given that X = 1, determine the conditional pmf of Yi.e., py|X(0I1), py|X(1|1), py|X(2|1). (Round your answers to four decimal places.) y 0 1 2 Ple(y|1) l l (b) Given that two hoses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island? (Round your answers to four decimal places.) y 0 1 2 PY|x(y|2) l W V 7 l V (c) Use the result of part (b) to calculate the conditional probability P(Y s 1 | X = 2). (Round your answer to four decimal places.) P(Y s 1 I x = 2) = l (d) Given that two hoses are in use at the full-service island, what is the conditional pmf of the number in use at the self-service island? (Round your answers to four decimal places.) x | o 1 2 poXIZ) | l l l l l . Need Help? i i An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the rst part and Y = the number of points earned on the second part. Suppose that the joint pmf ofX and Y is given in the accompanying table. y 0 0.03 0.06 0.02 0.10 x 5 0.04 0.13 0.20 0.10 10 0.01 0.15 0.15 0.01 (a) If the score recorded in the grade book Is the total number of point: earned on the two parts, what Is the expected recorded score E(X + Y)? (Enter your answer to one declmal place.) (b) If the maximum of the two scores is recorded, what is the expected recorded score? (Enter your answer to two decimal places.) Need Help