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

D 4 are x and y independently distributed explain your

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Unformatted text preview: y ) fY ( y ) 2 3 (1 + 2xy ) 2 3 (1 + y ) 1 + 2xy , 1+y 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. (d) (4%) Are X and Y independently distributed? Explain your answer. Sol) X and Y are not independent since y appears in fX |Y (x|y ) and cannot be cancelled. This implies fX |Y (x|y ) = fX (x). 4. Let X and Y be independent random variables with densities fX (x) = √ c1 xe−2x , fY ( y ) = c2 y 2 e−2y , x>0 y>0 (a) (6%) Find constants c1 and c2 . Sol) A gamma distributed random variable, say, U ∼ Γ(α, λ) has pdf: fU (u) = λα α−1 −λu u e , Γ(α) u>0 By comparing fX and fY with fU , we can associate X and Y with corresponding gamma distributions. So, X ∼ Γ(3/2, 2) and Y ∼ Γ(3, 2). These imply fX (x) = fY ( y ) = 23/2 √ −2x 25/2 √ −2x xe =√ xe , x>0 Γ(3/2) π 23 2 −2y ye = 2 2 y 2 e − 2y , y > 0 Γ(3) √ Accordingly, we have c1 = 25/2 / π and c2 = 22...
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This document was uploaded on 11/01/2013.

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