$X$ is a random variable with $f\_X (x) = 3x^2$ on the interval $(0, 1)$ and $Y$ is a random variable with conditional density $f\_{Y\mid X}(y\mid x) = \frac 1 x$
when $0 < y < x$ and $0 < x < 1$.

Describe the conditional distribution of $Y$ given $X = x$, and without calculation determine $E(Y \mid X = x)$.

Update:

I have since worked out that for a fixed x , the conditional density of y given x is constant and the distribution is therefore a Uniform Distribution. I still do not know how to go about the $E(Y \mid X = x)$ without calculation though!