Question: 9. Let X and Y have joint density function f (x, y) = ( 1 4 (x + 3y)e(x+y) if x, y 0, 0

9. Let X and Y have joint density function f (x, y) =

(

1 4 (x + 3y)e−(x+y) if x, y ≥ 0, 0 otherwise.

Find the marginal density function of Y. Show that P(Y > X) = 5 8 .

10. Let Sn be the sum of n independent, identically distributed random variables having the exponential distribution with parameter λ. Show that Sn has the gamma distribution with parameters n and λ.

For given t > 0, show that Nt = max{n : Sn ≤ t} has a Poisson distribution.

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