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# rec09-1 - | X> t where t ≥ 0 and z ≥ 0 3 Problem 3.7...

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Massachusetts Institute of Technology 6.041/6.431: Probabilistic Systems Analysis (Fall 2008) Recitation 9 October 2, 2008 1. Problem 3.1, page 184 of the text. Let X be uniformly distributed in the unit interval [0 , 1]. Consider the random variable Y = g ( X ), where g ( x )= ( 1 , if x 1 / 3; 2 , if x> 1 / 3 . Find the expected value of Y by ±rst deriving its PMF. Verify the result using the expected value rule. 2. The random variable X is exponentially distributed with parameter λ , i.e., f X ( x )= ( λe λx ,x 0; 0 , otherwise . (a) Calculate E [ X ], var( X ) and ±nd P ( X E [ X ]). Hint: P ( X z )= Z z f X ( x ) dx (b) Find P ( X t + z |
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Unformatted text preview: | X > t ), where t ≥ 0 and z ≥ 0. 3. Problem 3.7, page 186 of the text. Alvin throws darts at a circular target of radius r and is equally likely to hit any point in the target. Let X be the distance of Alvin’s hit from the center. (a) Find the PDF, the mean, and the variance of X . (b) The target has a concentric inner circle of radius t , with t < r . If X ≤ t , Alvin gets a score of S = 1 /X . Otherwise his score is S = 0. Find the CDF of S . Is S a continuous random variable? Compiled September 29, 2008 Page 1 of 1...
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