University of California, Berkeley - CS 70 dis6B- Solution. CS 70 Discrete Mathematics and Probability Theory Summer 2020. Questions and Answers.

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CS 70 Discrete Mathematics and Probability Theory Summer 2020 Course Notes DIS 6B 1 Condition on an Event The random variable X has the PDF fX(x) = (cx 0;-2; if 1 otherwise ≤ x ≤:2; (a) Deter... mine the value of c. (b) Let A be the event fX > 1:5g. Calculate P(A) and the conditional PDF of X given that A has occurred. 2 Max of Uniforms Let X1;:::Xn be independent U[0;1] random variables, and let X = max(X1;:::Xn). Compute each of the following in terms of n. (a) What is the cdf of X? (b) What is the pdf of X? CS 70, Summer 2020, DIS 6B 1 (c) What is E[X]? (d) What is Var[X]? 3 Darts but with ML Suppose Alice and Bob are playing darts on a circular board with radius 1. When Alice throws a dart, the distance of the dart from the center is uniform [0;1]. When Bob throws the dart, the location of the dart is uniform over the whole board. Let X the a random variable corresponding to the distance of the player’s dart from the board. (a) What is the pdf of X if Alice throws (b) What is the pdf of X if Bob throws (c) Suppose we let Alice throw the dart with probability p, and let Bob throw otherwise. What is the pdf of X (your answer should be in terms of p)? (d) Using the same premise as in part c, suppose you observe a dart on the board but don’t know who threw it. Let x be the dart’s distance from the center. We would like to come up with a decision rule to determine whether Alice or Bob is more likely to have thrown the dart given your observation, x. Specifically, if we let A be the event that Alice threw the dart and B be the CS 70, Summer 2020, DIS 6B 2 event that Bob threw, we want to guess A if P[AjX 2 [x;x + dx]] > P[BjX 2 [x;x + dx]] (what do these two probabilities have to sum up to?). For what values of x would we guess A? (your answer should be in terms of p) [Show More]

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