Paper 2, Section I, F

Probability | Part IA, 2017

Let XX and YY be real-valued random variables with joint density function

f(x,y)={xex(y+1) if x0 and y00 otherwise. 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 YY given XX.

(ii) Find the expectation of YY given XX.

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