rec10-1 - t time units of operation(b Find the PDF of the...

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Massachusetts Institute of Technology 6.041/6.431: Probabilistic Systems Analysis (Fall 2008) Recitation 10 October 7, 2008 1. Let X and Y be two random variables having the following joint PDF: f X,Y ( x, y ) = b cxy 2 , for y [0 , 1] , x [0 , 2 - 2 y ]; 0 , otherwise . (a) What is the value of c ? (b) What is the marginal PDF of Y , f Y ( y )? (c) What is the conditional PDF of X given Y , f X | Y ( x | y )? (d) Are X and Y independent? 2. Any computer chip will eventually fail, creating a random lifetime. Suppose that a manufac- turing process produces a mix of “good” and “bad” chips. For some positive number α , the lifetimes of good chips have the exponential distribution with parameter α and the lifetimes of bad chips have the exponential distribution with parameter 1000 α . Assume that the fraction of good chips is p and the fraction of bad chips 1 - p . (a) Find the probability that a randomly selected chip is still functioning after
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Unformatted text preview: t time units of operation. (b) Find the PDF of the time of failure of a randomly selected chip. (c) To weed out bad chips, each chip is tested for t time units, and only chips that do not fail during the testing period are shipped to customers. Find a formula for the probability that a customer receives a bad chip (as a function of the constants α , p , and t ). If p = 0 . 9, how long should the testing be to make the probability of shipping bad product be below 1%? 3. Suppose X 1 , X 2 , . . . , X n are independent and identically distributed (i.i.d.) random variables with the uniform distribution over [0 , 1]. (a) Let Y = max( X 1 , X 2 , . . . , X n ). Find the CDF of Y . (b) Let Z = min( X 1 , X 2 , . . . , X n ). Find the CDF of Z . (c) Find the joint CDF of random variables Y and Z . Compiled October 6, 2008 Page 1 of 1...
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This note was uploaded on 12/02/2011 for the course ENGINEERIN EE302 taught by Professor Proasin during the Spring '11 term at South Carolina.

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