Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Recitation 9 October 7, 2010 1. Let X be an exponential random variable with parameter λ > 0. Calculate the probability that X belongs to one of the intervals [n, n + 1] with n odd. 2. (Example 3.13 of the text book, page 165) Exponential Random Variable is Memoryless. The time T until a new light bulb burn s out is an exponential random variable with parameter λ. Ariadne turn s the light on, leaves the room, and when she r etur ns, t time units later, finds that the bulb is still on, which corresponds to the event A = {T > t}. Let X be the additional time until the bulb burns out. What is th e conditional CDF of X, given the event A? 3. Problem 3.23, page 191 in th e text. Let the random variables X and Y have a joint PDF which is uniform over the triangle with vertices (0, 0), (0, 1), and (1, 0). (a) Find the joint PDF of X and Y . (b) Find the marginal PDF of Y . (c) Find the conditional PDF of X given Y . (d) Find E[X | Y = y], and use the total expectation theorem to find E[X] in terms of E[Y ]. (e) Use the symmetry of the problem to find the value of E[X]. 4. We have a stick of unit length, and we break it into three pieces. We choose randomly and independently two points on the stick using a uniform PDF, and we br eak the stick at these points. What is the probability that the three pieces we are left with can form a triangle? Textbook problems are courtesy of Athena Scientific, and are used with permission. Page 1 of 1MIT OpenCourseWare http://ocw.mit.edu 6.041 / 6.431 Probabilistic Systems Analysis and Applied Probability Fall 2010 For information about citing these materials or our Terms of Use, visit:
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