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University of California, Berkeley - CS 70 dis13a CS 70 Discrete Mathematics and Probability Theory Spring 2017- All Solutions Worked.

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CS 70 Discrete Mathematics and Probability Theory Spring 2017 Rao DIS 13a 1 Uniform Means Let X1;X2;:::;Xn be n independent and identically distributed uniform random variables on the interval ... [0;1]. (a) Let Y = minfX1;X2;:::;Xng. Find E(Y). [Hint: Use the tail sum formula, which says the expected value of a nonnegative random variable is E(X) = R0¥ Pr(X > x)dx. Note that we can use the tail sum formula since Y ≥ 0.] (b) Let Z = maxfX1;X2;:::;Xng. Find E(Z). [Hint: Find the CDF.] 2 Conditioning on Exponentials Let Xi be i.i.d. Expo(l) random variables. (a) Compute E[Y j Z], where Y = maxfX1;X2g and Z = minfX1;X2g. (b) Compute E[X1 +X2 j Z]. [Hint: Use part (a).] (c) Use part (b) to compute E[Z]. (d) Compute E[X1 +X2 j X1 +X2 +X3]. CS 70, Spring 2017, DIS 13a 3 3 Bayesian Darts You play a game of darts with your friend. You are better than he is, and the distances of your darts to the center of the target are i.i.d. U[0;1] whereas his are i.i.d. U[0;2]. To make the game fair, you agree that you will throw one dart and he will throw two darts. The dart closest to the center wins the game. What is the probability that you will win? Note: The distances from the center of the board are uniform. [Show More]

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