math170bhw1 - p of heads repeatedly. Let N be a random...

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Mathematics 170B – HW1 – Due Thursday, January 6, 2011. Problems 1, 2, 3, 4 on page 246. (Note: On problem 4, the PDF of X is general – not the uniform from Example 3.14 in Chapter 3.) The following problems are from my 170A ±nal exam last Fall. They are the ones on which my students had the most di²culty. I am making them part of the ±rst 170B assignment to serve as review of 170A. A. A coin with probability p of heads is tossed until the ±rst head occurs. It is then tossed again until the ±rst tail occurs. Let X be the total number of tosses required. (a) Find the PMF of X . (b) Find the mean and variance of X . B. Suppose X has the N (0 , 1) distribution. Find the PDF of Y = 1 /X 2 . C. Toss a a biased coin with probability
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Unformatted text preview: p of heads repeatedly. Let N be a random variable that is independent of the tosses, and has a Poisson distribution with parameter λ . Let X be the number of heads obtained in the ±rst N tosses. What is the distribution of X ? (You should do a computation, not just give an answer.) D. Suppose X and Y are independent random variables with the exponential distribution with parameter 1. Let U = max( X,Y ), V = min( X,Y ) and W = U-V . (a) Compute P ( U ≤ u,V ≥ v ) for 0 ≤ v ≤ u . (b) Compute the joint PDF f ( u,v ) of ( U,V ). (c) Compute the CDF of W . (d) What is the distribution of W ? 1...
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This note was uploaded on 01/07/2011 for the course MATH 170b taught by Professor Staff during the Winter '08 term at UCLA.

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