Finding marginal distributions when y depends on x

[Gauss M.D.]

Homework Statement

(X,Y) is uniformly distributed over the area

T = {(x,y): 0 < x < 2, -x < 2y < 0}

Find the marginal probability functions ie $f_{x}(x)$ and $f_{y}(y)$.

The Attempt at a Solution

The thing I'm having trouble with is that y depends on x. Am I supposed to rewrite the boundaries for each marginal function? It feels like I'm doing things a roundabout way!

F(x,y) = $\int\int dx dy$

I.e. to find f(y):

-x < 2y < 0 $\Leftrightarrow$ x > -2y > 0 $\Rightarrow$ -2y < x < 2

Which means I can integrate with respect to x from -2y to 2, leaving me with f(y) = 2 + 2y

And if I instead want to find f(x):

-x < 2y < 0 $\Leftrightarrow$ -(1/2)x < y < 0

Which means I integrate with respect to y from -(1/2)x to 0, leaving me with f(x) = (1/2)x.

Again, it feels pretty roundabout and I wanted to make sure I wasn't missing anything.

 

[Gauss M.D.]

Also, if I wanted to find E(X) and E(Y) here after finding f(x) and f(y), what interval should I integrate x*f(x) over, given that they are bounded by each other?

 

[haruspex]

Your method of extracting f(x) and f(y) looks fine, and gets the right answers.
For E(X), just integrate xf(x) over the full range of x, etc.

 

[Gauss M.D.]

Thanks Haruspex!