Joint probability distribution. Please help.

F(x, y) = 1 - e^-x - e^-y + e^(-x-y), x>0, y>0. I am supposed to find P(X + Y > 3).

I am messing up somewhere along the way, and I don't know where. What I'm doing is:

1. Finding f(x, y) = F_xy(x, y), which comes to e^-x * e^-y.

2. Setting P(X + Y > 3) = 1 - P(X + Y <= 3).

3. Setting P(X + Y <= 3) = $\displaystyle \int_0^3 \int_0^{3-x} {e^{-x} e^{-y}} dy dx$

4. Solving to get 4e^-3. But the back of the book says the answer is (e^-2 - e^-3)^2. (Headbang)

I really have no clue what I'm doing wrong. The book is very unclear with its examples; it only gives rectangular regions.

Please help. I really need to get this done. And I am still awaiting help on this problem, which I am no closer to solving.

Re: Joint probability distribution. Please help.

Hey phys251.

According to Wolfram Alpha, your integral is correct (when you take 1 - value of integral).

int e^(-x)*e^(-y)dydx, y = 0 to 3 - x, x = 0 to 3, - Wolfram|Alpha

Its always a good idea to check with Wolfram if you need another independent opinion.

Re: Joint probability distribution. Please help.

Ha! So the back of the book is wrong?

Re: Joint probability distribution. Please help.

If the limits and the PDF are right, then yes the book is wrong.

The PDF looks right and so do the limits so I tend to agree with your answer.

If you need further confidence in your answer, check other sources or post the question in another area (forum, lecturers office, TA, other friend, etc).