(20 points) Rolling a fair n-sided die gives us a random number uniformly distributed among {1,...

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(20 points) Rolling a fair n-sided die gives us a random number uniformly distributed among {1, 2,...,n} (our usual die is n = 6). Suppose we independently roll a fair n-sided die twice and obtain numbers X1 and X2. (a) Calculate E(max(X1, X2)) and E(min(X1, X2)). (b) Show that your calculation in part (a) verifies that E(max(X1, X2)) + E(min(X1, X2)) = E(X1) + E(X2). (c) Is the above equation simply a coincidence? Can you give a simple and direct proof of it? (20 points) Rolling a fair n-sided die gives us a random number uniformly distributed among {1, 2,...,n} (our usual die is n = 6). Suppose we independently roll a fair n-sided die twice and obtain numbers X1 and X2. (a) Calculate E(max(X1, X2)) and E(min(X1, X2)). (b) Show that your calculation in part (a) verifies that E(max(X1, X2)) + E(min(X1, X2)) = E(X1) + E(X2). (c) Is the above equation simply a coincidence? Can you give a simple and direct proof of it?

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a To calculate EmaxX1X2 and EminX1X2 we need to consider all the possible outcomes and their probabi... View the full answer