Solutions - 5220-06

(a) Range of $W$ is $[\mathrm{0,1}]$.

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(b)

$$\begin{array}{ccc}& F_W(w)=P[Y-X\le w]\gg \gg P[Y\le w+X]& \\ & & \\ & \gg \gg 1-{\int }_0^{1-w}{\int }_{w+x}^{1}8xydydx& \\ & & \\ & \gg \gg \frac{8}{3}w-2w^2+\frac{1}{3}{w}^4& \text{(}w\in [\mathrm{0,1}]\text{)}\end{array}$$

Alternate:

$$F_W(w)={\int }_0^{1-w}{\int }_x^{w+x}8xydydx+{\int }_{1-w}^1{\int }_x^{1}8xydydx$$

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(c)

$$P[Y-X\ge 2/3]\gg \gg 1-F_W(2/3)\gg \gg 11/243$$