416-2 - (b) Find the expected length of the second run in...

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Denker Spring 2010 416 Stochastic Modeling - Assignment 2 Due Date: Monday, Feb. 1, 2010 Problem 1: (Exercise 11, p. 166) The joint density of X and Y is f ( x, y ) = 1 8 ( y 2 - x 2 ) e - y , 0 < y < , - y x y. Explain what E [ X | Y = y ] means and show that E [ X | Y = y ] = 0 holds for any positive y . Problem 2: (Exercise 26, p.169) You have two opponents, A and B, with whom you alternate play. When you play A, you win with probability p A ; whenever you play B, you win with probability p B , where p B < p A . If your objective is to minimize the number of games you need to play to win two in a row, should you start with A or B? First give an intuitive reasoning what answer you expect, then give a proof. Problem 3: (Exercise 13, p.166) Let X be exponential with mean λ ; that is f X ( x ) = 1 λ e - x/λ 0 < x < . Explain the notion E [ X | X > 1] and ±nd its value. Problem 4: (Exercise 31, p. 170) Each element in a sequence of binary data is either 1 with probability p or 0 with probability 1 - p . A maximal subsequence of consecutive values having identical outcomes is called a run. (a) Find the expected length of the ±rst run in the binary sequence.
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Unformatted text preview: (b) Find the expected length of the second run in the binary sequence. Problem 5: (Exercise 21, p. 168) A miner is trapped in a mine containing three doors (possible exits). The frst leads to a tunnel that takes him to saFety aFter two hours oF travel. The second door leads to a tunnel that returns him to the mine aFter fve hours oF travel. The third dooor leads to another tunnel that returns him to the mine aFter three hours. The miner chooses at each time any oF the doors at random. Let N be the total number oF doors selected beFore the miner reaches saFety. Let T i denote the travel time corresponding to the i th choice oF the door ( i ≥ 1). ±inally, let X denote the time until the miner reaches saFety. (a) ±ind an identity that relates X to N and the T i . (b) ±ind EN . (c) ±ind E ( T N ). (d) ±ind E [ ∑ N i =1 T i | N = n ]. (e) Using the preceding, what is EX ?...
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This note was uploaded on 04/15/2010 for the course STAT 416 taught by Professor Denker during the Spring '08 term at Penn State.

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416-2 - (b) Find the expected length of the second run in...

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