Let $X$ and $Y$ be real-valued random variables with joint density function

$$f(x, y)= \begin{cases}x e^{-x(y+1)} & \text { if } x \geqslant 0 \text { and } y \geqslant 0 \\ 0 & \text { otherwise. }\end{cases}$$

(i) Find the conditional probability density function of $Y$ given $X$.

(ii) Find the expectation of $Y$ given $X$.