bounds for conditional expectation and variance

Question

let X and Y be random variables with density:

$$f_{x,y}(x,y)=\frac{1}{\pi}\mathbf{1}_{\{x^2+y^2\le1\}}$$

Find $$E[X|Y] \& Var(X|Y)$$

So what I did was:

$$E[X|Y]=\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\frac{x}{2\sqrt{1-y^2}}dx=0$$

$$Var(X|Y)=\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\frac{x^2}{2\sqrt{1-y^2}}dx=-\frac{y^2-1}{3\sqrt{1-y^2}}$$

Does this seem right? I'm not sure about the bounds I used to get the marginal distributions.

Answer 1

The variance is wrong. But without a lot of calculations, when you got your conditional density

$f_{X|Y}(x|y)=\frac{1}{2\sqrt{1-y^2}}$

$-1\leq y \leq 1$

You can observe that this is a Uniform$(a;b)$ distribution

$$(X|Y)\sim U(-\sqrt{1-y^2};\sqrt{1-y^2})$$

Then you can calculate

• mean as: $\frac{a+b}{2}=0$

• variance as: $\frac{(b-a)^2}{12}=\frac{1-y^2}{3}$