- #1
Gauss M.D.
- 153
- 1
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 [itex]f_{x}(x)[/itex] and [itex]f_{y}(y)[/itex].
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) = [itex]\int\int dx dy[/itex]
I.e. to find f(y):
-x < 2y < 0 [itex]\Leftrightarrow[/itex] x > -2y > 0 [itex]\Rightarrow[/itex] -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 [itex]\Leftrightarrow[/itex] -(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.