Finding the conditional expectation of $X$ and $Y$ with $f(x,y) = c\cos x$ when $0<y<x<\frac{\pi}{2}$

Question

$X,Y$ are random variables with joint density: $$f(x,y) = \begin{cases} c\cos x,& 0<y<x<\frac{\pi}{2}\\ 0,& \text{otherwise.} \end{cases}$$

I found the $ f_{Y|X=x} = 1$ and solved $ E(Y|X=x) = \int_0^x y\cdot1 dy = x^2/2 $, but the book gives the answer $x/2$

Could you please explain in detail where I went wrong?

Thank you very much!

Answer 1

Just so this does not get added to the list of unanswered questions....

It is true that the conditional density of $Y$ given that $X = x$ is a uniform density (you don't even need to find the numerical value of $c$ to arrive at this conclusion), but, given that $X = x$, $Y$ can take on values in $(0,x)$ only, and so the conditional density of $Y$ given that $X=x$ has value $\frac 1x$ for $y \in (0,x)$, and not value $1$ as you claim to have found. Hence, $E[Y\mid X = x] = \frac x2$ as your book states.