Jointly continuous random variables

[DotKite]

Homework Statement

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.

Homework Equations

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

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.

 

[DotKite]

Never mind I figured out my mistake. The limits of the first integral should not be 0 to infinity. They are 0 to 1

 

[Ray Vickson]

Never mind I figured out my mistake. The limits of the first integral should not be 0 to infinity. They are 0 to 1

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

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