OK, let's see if we can find where my rusty calculus skills betrayed me.

(going to avoid using latex to put in the integral since it seems to come out weird when I preview it)

joint density = e

^{-x1}e

^{-x2}
we want: integral(0<=X

_{2}<=y) of e

^{-x2} times

**(** integral(0<=X

_{1}<=y-x

_{2}) of e

^{-x1}dx

_{1} **)** times dx

_{2}
that resolves to: integral(0<=X

_{2}<=y) of e

^{-x2} times

**(** -e

^{-t} **)**^{t=y-x2}_{t=0} times dx

_{2}
which resolves to: integral(0<=X

_{2}<=y) of e

^{-x2}(1-e

^{y}e

^{-x2})dx

_{2}
we can distribute that out to:

**(** integral(0<=X

_{2}<=y) of e

^{-x2}dx

_{2} **)** -

**(** e

^{y} * integral(0<=X

_{2}<=y) of e

^{-x2}e

^{-x2}dx

_{2} **)**
after we integrate we get:

**(** -e

^{-t} **)**^{t=y}_{t=0} - e

^{y}**(** (-1/2)e

^{-2t} **)**^{t=y}_{t=0}
which gives: 1-e

^{-y} + (1/2)e

^{y}(1-e

^{-2y})

a little quick distribution turns that into: 1 - e

^{-y} + (1/2)(e

^{y}-e

^{-y})

which finally gives us: 1 + (1/2)e

^{y} - (3/2)e

^{-y}
...not what we're supposed to get.