Conditional PDF of this random variable

[nacho-man]

the random variable X and Y have a joint PDF given by:

$f_{x,y}(x,y) = \frac{1}{10}$, $(x,y)\in[-1,1] * [-2,2] \cup [1,2] * [-1,1]$

a) find the conditional PDF for $f_{y|x}(x,y)$ and $f_{x|y}(xy)$

and
b) find E[X|Y], E[X] and Var[X|Y]. Use these to calculate var(X)

for part a) I am unsure how to interpret the given domain, and how to use it to find the PDF.

Could I get some guidance?

for
a) i got $f_{X|Y}(x,y) = 1/3$ and $f_{Y|X}(x,y) = 1/4$ over the appropriate domains/ranges

 

[Jhero]

.for b) I got $E[X|Y] = 2/3$, $E[X] = 1/2$ and $Var[X|Y] = 1/12$. So, $Var[X] = \frac{1}{12} + \frac{4}{9} - \frac{4}{3} = \frac{7}{36}$.