rec08 - 5 minutes Find the CDF and the expected value of...

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Massachusetts Institute of Technology 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Recitation 8 October 5, 2010 1. Let Z be a continuous random variable with probability density function f z ( z ) = b γ (1 + z 2 ) , if - 2 < z < 1 , 0 , otherwise . (a) For what value of γ is this possible? (b) Find the cumulative distribution function of Z . 2. Problem 3.9, pages 186–187 in the text. The taxi stand and the bus stop near Al’s home are in the same location. Al goes there at a given time and if a taxi is waiting, (this happens with probability 2/3) he boards it. Otherwise he waits for a taxi or a bus to come, whichever comes ±rst. The next taxi will arrive in a time that is uniformly distributed between 0 and 10 minutes, while the next bus will arrive in exactly
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Unformatted text preview: 5 minutes. Find the CDF and the expected value of Al’s waiting time. 3. Let λ be a positive number. The continuous random variable X is called exponential with parameter λ when its probability density function is f X ( x ) = b λe-λx , if x ≥ , , otherwise . (a) Find the cumulative distribution function (CDF) of X . (b) Find the mean of X . (c) Find the variance of X . (d) Suppose X 1 , X 2 , and X 3 are independent exponential random variables, each with param-eter λ . Find the PDF of Z = max { X 1 ,X 2 ,X 3 } . (e) Find the PDF of W = min { X 1 ,X 2 } . Page 1 of 1...
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