Hints for Homework Assignment 10

• Problem 1: (a) through (c) are routine; for (d) recall how one can deal with an average (or sum) of a large number of independent random variables ...
• Problem 2: (a) and (b) reduce to the computation of integrals. Some care has to be taken to deal with the absolute value: split the integrals into two parts, corresponding to the ranges x>0 and x<0. The computation of the expectation and variance is a bit tedious as it requires integration by parts (in the case of the variance, two integrations by parts);
• Problem 3: A routine Poisson process problem.
• Problem 4: For each part draw the regions (inside the unite square) corresponding to the given probabilities. Because of the independence and uniform distribution of X and Y, the probabilities are proportional to the areas of these regions.
• Problem 5: (a) - (c) are simple integration exercises.
• Problem 6: Use the properties of expectation!
• Problem 7: The theoretical version (part (a)) boils down to the computation of a double integral.

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