Quiz2-2007-2- test 2- 610 question 1

Therefore fz z 292 72 2z ze 92 z0 5 4 let

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Unformatted text preview: . 2 (b) (4%) Find the density of the random variable Z = X + Y . Sol) From theorem for the sums of gamma distributed random variables, we immediately have Z ∼ Γ(9/2, 2). Therefore, fZ (z ) = 29/2 7/2 −2z ze, Γ(9/2) z>0 5. (4%) Let X and Y be jointly continuous with joint probability density function given by 1 for 1 < x < ∞ and 1 < y < ∞ x2 y 2 f (x, y ) = 0 otherwise. Find the probability density function of U = XY . Sol) It is clear that I (U ) = (1, ∞). Consider u ∈ I (U ). We have ∞ fU (u) = = = u 1 f x, dx |x| x −∞ u 11 · dx x u2 1 ln u , 1 < u < ∞. u2 6. Suppose that the time interval between arrivals of busses to a particular stop is a random variable X (in hours) with exponential distribution and parameter λ = 2 (arrivals/hour). (a) (4%) You arrive, panting, to t...
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