Solutions - 5250-02

(a)

Obtain the joint from the conditional and marginal:

$$\begin{array}{c}{f}_{X,Y}(x,y)=f_{Y|X}(y|x)\cdot f_X(x)\\ \\ \gg \gg 3x^2\cdot \frac{2y}{x^2}\gg \gg 6y\end{array}$$ $$f_{X,Y}(x,y)=\{\begin{array}{cc}6y& 0\le y\le x\le 1\\ 0& \text{otherwise}\end{array}$$

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(b)

(1) Compute the marginal $f_Y(y)$:

$$\begin{array}{c}{f}_Y(y)={\int }_y^{1}6ydx\\ \\ \gg \gg {6yx|}_y^1\gg \gg 6y(1-y)\end{array}$$

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(2) Apply the conditional density formula:

$$\begin{array}{c}{f}_{X|Y}(x|y)=\frac{f_{X,Y}(x,y)}{f_Y(y)}\\ \\ \gg \gg \frac{6y}{6y(1-y)}\gg \gg \frac{1}{1-y}\end{array}$$ $$f_{X|Y}(x|y)=\{\begin{array}{cc}{\displaystyle \frac{1}{1-y}}& 0\le y\le x\le 1\\ 0& \text{otherwise}\end{array}$$