Cumulative distribution and probability density functions of the random variable

Question

Suppose that $X \sim U(0,1)$. Determine cumulative distribution and probability density functions of the random variable $^$

Taking my first probability class and it's been giving me some pretty hard times so far. Is this the right approach to the problem? Thanks in advance.

Let $Y = e^x$

$F_y(y) = P(Y ≤ y) = P(e^x ≤ y)$

$=P(ln(e^x) ≤ ln(y))$

$ = P(x ≤ ln(y)) = F_x(ln(y)) = \frac{ln(y)}{1-0} = ln(y)$, if $0 < x < 1$

So if $e < y < e^2$, then $fy(y)=F_y = \frac{1}{y}$ and $fy(y) = 0$ otherwise

Answer 1

Almost correct. Note that $$ F_X(\ln y) = \begin{cases} 0, & \ln y \le 0 \iff y \le 1\\ \ln y, & 0 < \ln y < 1 \iff 1 <y < e\\ 1, & \text{otherwise} \end{cases} $$

Please apply to your result.