Question

Suppose we conduct a study of heights of fathers and their sons in a particular population, letting X be the fathers height in inches and Y the sons. Further, suppose that the random pair (X,Y) is distributed as bivariate normal with EIX) = EY] 68, Var(X) = Var(y) = 4, Cov(X, y) = 06. In what follows, give explicit expressions and simplify them as much as possible. Show your work, not just the final answer. (a) What is the probability that the father is taller than the son? (b) What is the probability that the father is at least 4 inches taller than the son? (c) What is the distribution of the heights of sons whose fathers are 74 inches tall? (d) Given that a father is 74 inches tall, find the probability that the son is taller than the father. e) One hundred father-son pairs are randomly sampled. Let X be the sample average for fathers and Y the sample average for sons. What is the joint distribution of (X,Y)? (f) What is the probability that the two sample averages are within 3 inches of each other?

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dia iven tha Cov(x, Y) 6 cov(Y) 339(- b) fiven fnches talles than can the probabftity 1 c) tiven. tohose fathes ove inches ta3.9) 1 n hl f) tiven the Sample a vesages a coith in 3 inches N N (0, 0 061) 100 : 01061

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