Solutions - 5190-11

(a)

Express CDF of $W$:

$$F_W(w)=P[W\le w]=P[\mathrm{Max}(X,Y)\le w]$$

Case 1, $0<w<1$:

$$F_W(w)={\int }_0^w{\int }_0^w\frac{1}{2}dydx\gg \gg \frac{1}{2}{w}^2$$

Case 2, $1\le w<2$:

$$F_W(w)={\int }_0^1{\int }_0^w\frac{1}{2}dydx\gg \gg \frac{1}{2}w$$

Complete CDF:

$$F_W(w)=\{\begin{array}{cc}0& w<0\\ \frac{1}{2}{w}^2& 0\le w<1\\ \frac{1}{2}w& 1\le w<2\\ 1& w\ge 2\end{array}$$

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(b)

Compute $P[Y-X>1.25]=P[Y>X+1.25]$:

$$\begin{array}{c}\gg \gg {\int }_0^{\frac{3}{4}}{\int }_{x+1.25}^2\frac{1}{2}dydx\\ \\ \gg \gg \frac{3}{4}\cdot \frac{3}{4}\cdot \frac{1}{2}\cdot \frac{1}{2}\gg \gg \frac{9}{64}\gg \gg 0.1406\end{array}$$