Conditional expectation of joint uniform

Question

Let $X,Y$ be random variables whose joint distribution is uniform on the triangle with vertices $(0,0), (1,0), (1,1)$. I wish to compute the conditional expectation $E[Y|X]$. How do I start?

Answer 1

$$f_{(X,Y)}(x,y) = 2 \, \, \, \mathbb{1}_{(A)}\hspace{.5cm} A = \{(x,y): 0 \leq y \leq x \leq 1 \}$$

$$f(y|x)=\frac{f_{(X,Y)}(x,y)}{f_{X}(x)} =\frac{2 \, \, \, \mathbb{1}_{(A)} }{\int f_{(X,Y)}(x,y) dy}$$

$$=\frac{2 \, \, \, \mathbb{1}_{(0 \leq y \leq x)} }{\int_0^x 2 dy} =\frac{2 \, \, \, \mathbb{1}_{(0 \leq y \leq x)} }{2x }=\frac{ \, \, \, \mathbb{1}_{(0 \leq y \leq x)} }{x }$$

$$E(Y|X=x)=\int y f(y|x) dy=\int y \frac{ \, \, \, \mathbb{1}_{(0 \leq y \leq x)} }{x } dy=\frac{ 1}{x }\int_0^x y dy=\frac{ x}{2 }$$

so $$E(Y|X)=\frac{ X}{2 }$$