Content Bb 21051217 C learn-us-east-1-prod-fleet02-xythos.content.blackboardcdn.com/5cc71db6522fe/21051217?X-Blackboard-Expiration=1634860800000&X-Blackboard-Signature=7706... To E 21051217 1 / 5 | - 100% + Show your calculations and explain your answers where needed. (1) Pick a hand of 5 cards at random from a standard 52-card deck. Let A be the number of Aces and K the number of kings in your hand. Let H be the event that all 5 cards are Hearts. 1) Calculate P(H). 2) Calculate E(A). 3) Calculate E( AJH). 1 Calculate E( AK). 5) Are K and A independent conditioned on H? 2 3Content Bb 21051217 C learn-us-east-1-prod-fleet02-xythos.content.blackboardcdn.com/5cc71db6522fe/21051217?X-Blackboard-Expiration=1634860800000&X-Blackboard-Signature=7706... * To E 21051217 2 / 5 100% + 2 (2) Let m be a fixed non-negative number and let M ~ Pois(m). Conditioned on M, place M letters uniformly and independently at random into n boxes. Let X; be the number of letters in box i. 1) Using the Law of Total Expectation, compute EX1. 2) Using the Law of Total Expectation, compute var X1. 3) Using the Law of Total Expectation, find the distribution of X1. (Hint: compute Pr(X1 = k) for a non-negative integer k). 4) Are the events {X1 = 0} and {X2 = 0} independent? 5) Are X1 and X2 independent random variables? 2 3Content Bb 21051217 C learn-us-east-1-prod-fleet02-xythos.content.blackboardcdn.com/5cc71db6522fe/21051217?X-Blackboard-Expiration=1634860800000&X-Blackboard-Signature=7706... * To E 21051217 3 / 5 | 100% + 3 (3) Let Z ~ N(0, 1) and let Xn ~ Unif[-1, 1]. Let Zn = Z + Xn. 1) Prove that Xn - 0. 2) Prove that Zn " Z. 1 2 3Content Bb 21051217 C learn-us-east-1-prod-fleet02-xythos.content.blackboardcdn.com/5cc71db6522fe/21051217?X-Blackboard-Expiration=1634860800000&X-Blackboard-Signature=7706... * To 21051217 4 / 5 100% + 4 (4) Let X = +1 with probability 1/2 each. Let Xn = (-1)" . X. 1) Prove that Xn = X. 2) Prove that Xn does not converge to X (or any other random variable) in proba- bility. 2 3Content Bb 21051217 C learn-us-east-1-prod-fleet02-xythos.content.blackboardcdn.com/5cc71db6522fe/21051217?X-Blackboard-Expiration=1634860800000&X-Blackboard-Signature=7706... To E 21051217 5 / 5 | - 100% + (5) Let Xn be a discrete uniform random variable on {1, ..., n}. Let Xn = Xn. Let U ~ Unif[0, 1]. 1) Prove that Xn = U. 2 3