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Homework Help: Jointly continuous random variables

  1. Mar 8, 2014 #1
    1. The problem statement, all variables and given/known data

    Let X and Y be random losses with joint density function

    f(x,y) = e^-(x + y) for x > 0 and y > 0 and 0 elsewhere

    An insurance policy is written to reimburse X + Y:
    Calculate the probability that the reimbursement is less than 1.

    2. Relevant equations

    Have not learned independence for jointly cont r.v's yet

    3. The attempt at a solution

    p(X + Y < 1) = p(Y < 1 - X) = ##\int_{0}^{\infty}\int_{0}^{1-x} e^{-(x+y)} dydx##

    When I go through solving this double integral I get the following

    ##-e^{-x} - xe^{-1}## evaluated from 0 to ∞.

    However as x → ∞ the above function diverges. Maybe I calculated the integral wrong, I have done it over and over, and cannot seem to find where it could be wrong.
  2. jcsd
  3. Mar 8, 2014 #2
    Never mind I figured out my mistake. The limits of the first integral should not be 0 to infinity. They are 0 to 1
  4. Mar 8, 2014 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    That was not your only mistake: you need to re-do the inner integral
    [tex] \int_0^{1-x} e^{-(x+y)} \, dy = e^{-x} \int_0^{1-x} e^{-y} \, dy. [/tex]

    Edit: Oh... maybe you wrote the indefinite x- integral of inner y-integral; in that case, you are correct.
    Last edited: Mar 8, 2014
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