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170B Probability TheoryPuck RombachHomework 1, due: 9/30/16.Problem 1Suppose that X and Y are independent, identically distributed, geometric random variables withparameter p. Show thatP(X = i|X > Y) > 2P(X = i)P(Y < i)……….Problem 2You have lost your keys, and you know they are in one of the rooms 1, …, n in your house. Theprobability of the keys being in any of those rooms is not a uniform distribution. The probabilityPthat the keys are in room i is pi (such that ni=1 pi = 1). Also, the rooms are not equally hard tosearch. If the keys are in room i and you search it, you will find them with probability qi . Given thatyou have searched room 1 and 2 unsuccessfully, find the pmf of the location of the keys……….Problem 3A stick of length L is broken into two pieces, such that the left piece has length X. The PDF of X isgiven by2xL2 , 0 ? x ? LfX (x) =0, otherwise.This process may then be repeated on one of the two pieces of the stick, where L is set to be thelength of that piece.(a) Check that the normalization property holds for this PDF.(b) Find E(X) and var(X)……….2Problem 4Let X be a continuous random variable with PDF:12 if 0 ? x ? 2,fX (x) =0 otherwise.Let Y = g(X) = ?X 2 .(a) What are RX and RY (the support of X and Y)?(b) Is g(X) strictly monotonic?(c) Find fY (y)……….170B Probability TheoryHomework Sheet 1