# Thread: Probability Question on jointly distributed function

1. ## Probability Question on jointly distributed function

I'm having problems figuring out the limits for this. The exponential is e^-(x + y) apologies about the lack of clarity on it.

2. Hello,

You must have $\displaystyle \iint_{\mathbb{R}^2} f_{X,Y}(x,y) ~dxdy=1$

That is to say, by writing the boundaries:

$\displaystyle \int_0^\infty \int_0^y cxe^{-(x+y)} ~dxdy=1$

maybe reversing the order of integration will help :
$\displaystyle =\int_0^\infty \int_x^\infty cxe^{-(x+y)} ~dydx=c\int_0^\infty x \int_x^\infty e^{-(x+y)} ~dydx=1$

and that's all

3. For part (c) here would the limits be 0,inf for y and 0, 2-y for x dxdy?

4. Originally Posted by DCU
For part (c) here would the limits be 0,inf for y and 0, 2-y for x dxdy?
integral of the pdf, yup that's correct

5. Thanks, I think I might pass this exam!

6. Originally Posted by DCU
Thanks, I think I might pass this exam!

and if you have any other question, you know where you can get answers/help...

7. I'd go one further and also pull out the $\displaystyle e^{-x}$, as I did below.

Originally Posted by Moo
Hello,

You must have $\displaystyle \iint_{\mathbb{R}^2} f_{X,Y}(x,y) ~dxdy=1$

That is to say, by writing the boundaries:

$\displaystyle \int_0^\infty \int_0^y cxe^{-(x+y)} ~dxdy=1$

maybe reversing the order of integration will help :
$\displaystyle 1=\int_0^\infty \int_x^\infty cxe^{-(x+y)} ~dydx=c\int_0^\infty x e^{-x}\int_x^\infty e^{-y} ~dydx$

and that's all
$\displaystyle 1=c\int_0^\infty x e^{-x}e^{-x} dx=c\int_0^\infty x e^{-2x}dx$

Which can either be solved by parts or by recognizing the constant in the gamma density
with $\displaystyle \alpha=2$ and what I call $\displaystyle \beta =.5$.