Solutions - 5150-05

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

$$\begin{array}{c}{F}_Y(y)=P[Y\le y]\\ \\ \gg \gg P[|X-2|\le y]\gg \gg P[-y<X-2<y]\\ \\ \gg \gg P[-y+2<X<y+2]\end{array}$$

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(2) Evaluate Case 1, $0<y<1$:

$$\begin{array}{c}{\int }_{-y+2}^{y+2}\frac{1}{3}dx\\ \\ \gg \gg \frac{1}{3}((y+2)-(-y+2))\gg \gg \frac{2y}{3}\end{array}$$

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(3) Evaluate Case 2, $1<y<2$:

$$\begin{array}{c}{\int }_1^{y+2}\frac{1}{3}dx\gg \gg \frac{1}{3}(y+1)\end{array}$$

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(4) State the complete CDF:

$$F_Y(y)=\{\begin{array}{cc}0& y<0\\ 2y/3& 0\le y<1\\ \frac{1}{3}(y+1)& 1\le y<2\\ 1& 2\le y\end{array}$$