Statistics Help: Conditional Variance

Question

I'm supposed to find the conditional variance.

My attempt: I'm doing so by first computing f(Y | X = 0.5).

I know this equals f(0.5, Y) / f(0.5)

But I'm having trouble figuring out what f(0.5) is? My plan is to use f(Y | X = 0.5) to find the conditional variance. Please help, thanks so much.

Answer 1

I'm having trouble figuring out what f(0.5) is?

It is : $$f_X(0.5)=\int_0^1 f_{X,Y}(0.5,y)\operatorname d y$$

My plan is to use f(Y | X = 0.5) to find the conditional variance.

Good plan.   So then:-

$$\begin{align}\mathsf E(Y^n\mid X\!=\!0.5) ~&=~ \dfrac{\int\limits_0^1 y^n~f_{X, Y}(0.5, y)\operatorname d y}{\int\limits_0^1 f_{X,Y}(0.5,y)\operatorname d y}\\[1.5ex] &=~ \dfrac{\int\limits_0^1 (0.5y^n+y^{n+1})\operatorname d y}{\int\limits_0^1 (0.5+y)\operatorname d y}\\[1ex]&=\lower{1ex}{\ldots\ldots\ldots}\\[3ex]\mathsf {Var}(Y\mid X\!=\!0.5)~&=~ \mathsf E(Y^2\mid X\!=\!0.5) -\mathsf E(Y\mid X\!=\!0.5)^2 \\[1ex]&=\lower{1ex}{\ldots\ldots\ldots}\end{align}$$