Solutions - 5150-06

(1) Express the CDF of $W$ in terms of $X$:

$$\begin{array}{c}{F}_W(w)=P[W\le w]\\ \\ \gg \gg P[3X-2\le w]\gg \gg P[X\le \frac{w+2}{3}]\end{array}$$

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(2) Evaluate Case 1, $-2\le w\le 1$:

$$\begin{array}{c}{\int }_0^{\frac{w+2}{3}}xdx\gg \gg {\frac{1}{2}{x}^2|}_0^{\frac{w+2}{3}}\\ \\ \gg \gg \frac{1}{2}{\left(\frac{w+2}{3}\right)}^2\gg \gg \frac{{(w+2)}^2}{18}\end{array}$$

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(3) Evaluate Case 2, $1\le w\le 2.5$:

$$\begin{array}{c}{\int }_0^{1}xdx+{\int }_1^{\frac{w+2}{3}}1dx\\ \\ \gg \gg {\frac{1}{2}{x}^2|}_0^1+{x|}_1^{\frac{w+2}{3}}\\ \\ \gg \gg \frac{1}{2}+\frac{w+2}{3}-1\gg \gg \frac{w+2}{3}-\frac{1}{2}\end{array}$$

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(4) State the complete CDF:

$$F_W(w)=\{\begin{array}{cc}0& w<-2\\ \frac{{(w+2)}^2}{18}& -2\le w<1\\ \frac{w+2}{3}-\frac{1}{2}& 1\le w<2.5\\ 1& 2.5\le w\end{array}$$