471inclass - X and Y(c Show that X and Y are independent...

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Prelim II, 50 minutes SHOW ALL WORK! (1) [30] Calculate the expectation and variance of a Gamma (7 ) random variable. Hint: you can do this any way you want, but one way is to use its relationship with other distributions. Be careful on the variance, if you use a theorem you should state it. Also, you may use the fact that e x grows faster than any polynomial. For problems (2) and (3) use the joint density f ( x,y ) = 4 81 xy on the square [0 , 3] × [0 , 3]. (2) [40 pts] (a) Find P ( XY > 3). (b) Find the marginal densities of
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Unformatted text preview: X and Y . (c) Show that X and Y are independent. Use any theorems you know. (3) [30 pts] Find the density f X + Y ( z ) of X + Y . Note that it is a piecewise function. You do not have to simplify expressions. (4) [0 pts] Would you prefer a take-home final due at noon on 12-5 (Monday of reading period) or an in-class final on 12-14 from 7-9:30pm. Say how strong your opinion is from 0 to 10 with 0 being for take-home, 10 being for in-class....
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