Solutions - 5210-01

(1) Find the CDF of $W=\ln X$:

$$\begin{array}{c}{F}_W(w)=P[W\le w]=P[\ln X\le w]=P[X\le e^w]\end{array}$$

Compute $F_X(x)$:

$$\begin{array}{c}{F}_X(x)={\int }_1^x\frac{2}{3}tdt\gg \gg {\left[\frac{t^2}{3}\right]}_1^x\gg \gg \frac{x^2-1}{3}\end{array}$$

Substituting $x=e^w$:

$$F_W(w)=\{\begin{array}{cc}0& w<0\\ \frac{e^{2w}-1}{3}& 0\le w\le \ln 2\\ 1& w>\ln 2\end{array}$$

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(2) Differentiate to find $f_W(w)$:

$$f_W(w)=\{\begin{array}{cc}0& w<0\\ \frac{2e^{2w}}{3}& 0\le w\le \ln 2\\ 0& w>\ln 2\end{array}$$