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Homework Help: Probability random vector, transformation

  1. Oct 8, 2011 #1
    1. The problem statement, all variables and given/known data

    Let f(x,y) = e^(-x-y), 0<x< infinity, 0<y<infinity, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute P(Z<=0), P(Z,<=6), and, more generally P(Z<=z), for 0<z<infinity. What is the pdf of Z?

    2. Relevant equations

    P(x,y) = ∫∫(f(x,y) dxdy

    3. The attempt at a solution

    so, P(Z<= 0 ) is pretty obviously 0

    P(Z<=6) = P(X+Y <= 6)

    into equation

    ∫from (0 to infinty) ∫ from (0 to 6-Y ) e^(-x-y) dx dy

    -e^(-x-y) eval from 0 to 6-Y
    = (-e^-6 ) + e^-y

    ∫from (0 to infinity) (-e^-6 ) + e^-y dy
    = -ye^(-6) -e^-y eval from (0 to infinity)
    = -infinity -0 - ( 0 - 1)
    = - infinity

    so, im definitly going wrong somewhere because a probability of negative infinity makes no sense... I know this could be done using a different method of tranformations, but i think im supposed to do something along these lines because thats what is taught in the preceeding chapter.
  2. jcsd
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