# Marginal distribution, double integral clarification

1. Apr 30, 2013

### Gauss M.D.

1. The problem statement, all variables and given/known data

(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 $f_{y}(y)$.

3. 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.

2. Apr 30, 2013

### 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?

3. Apr 30, 2013

### 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.

4. Apr 30, 2013

### Gauss M.D.

Thanks Haruspex!