Solutions - 5190-10

(a)

Marginal PDF of $X$:

$$f_X(x)=\{\begin{array}{cc}1-\frac{1}{2}x& 0\le x\le 2\\ 0& \text{otherwise}\end{array}$$

Then:

$$P[X>1.5]\gg \gg {\int }_{1.5}^{2.0}{f}_X(x)dx\gg \gg 0.0625$$

---

(b)

$$\begin{array}{c}\gg \gg {\int }_0^{1.5}{\int }_{x+\frac{1}{2}}^2\frac{1}{2}dydx\\ \\ \gg \gg \frac{1}{2}(1.5)(1.5)\cdot \frac{1}{2}\gg \gg 9/16\end{array}$$

---

(c)

$$\begin{array}{c}{F}_W(w)=P[W\le w]\gg \gg P[Y-X\le w]\\ \\ \gg \gg 1-{\int }_0^{2-w}{\int }_{w+x}^2\frac{1}{2}dydx\\ \\ \gg \gg w-\frac{w^2}{4}\end{array}$$ $$F_W(w)=\{\begin{array}{cc}0& w<0\\ w-\frac{w^2}{4}& 0\le w<2\\ 1& w\ge 2\end{array}$$