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Stat 5101, Fall 1999 (Geyer) Second Midterm Solutions

# Stat 5101, Fall 1999 (Geyer) Second Midterm Solutions - Up...

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Up: Stat 5101 Stat 5101 (Geyer) Midterm 2 Problem 1 This is a multivariate change of variable problem (Section 12.1 in Lindgren). The first step is to solve for the old variables in terms of the new variables. Clearly X = U V . Then Y = U - X = U (1 - V ). The next step is to determine the Jacobian of the transformation Clearly U = X Y can be any positive number. Given U , can be any number between 0 and 1 because X can be any number between X and U . Thus the joint density is Problem 2 This is a job for the ``recognizing the unnormalized density'' trick. First

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If we recognize the integrand as a density, we see the integral must give No further simplification is possible. The recursion formula for gamma functions only allows cancellation of gamma functions with arguments differing by an integer. Problem 3 (a) Let X be the random variable in question. The count of points occurring in a fixed time interval of length t for a Poisson process with rate parameter is . Here per day and t = 7 days. Thus the answer is .
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