i was wondering how to get density function from a mass function for another density function

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$W=e^{-Y}$ takes values in $\left(0,e^{-3}\right)$ so that $F_{W}\left(w\right)=1$ if $W\geq e^{-3}$ and $F_{W}\left(w\right)=0$ if $W\leq0$.

Consequently we can go for $f_W(w)=0$ for $w\notin\left(0,e^{-3}\right)$.

For $w\in\left(0,e^{-3}\right)$ we find $$F_{W}\left(w\right)=P\left(e^{-Y}\leq w\right)=P\left(-Y\leq\ln w\right)=P\left(Y\geq-\ln w\right)=1-F_{Y}\left(-\ln w\right)$$

Taking the derivative we find: $$f_{W}\left(w\right)=-f\left(-\ln w\right)\frac{d\left(-\ln w\right)}{dw}=\frac{f\left(-\ln w\right)}{w}=\frac{e^{\ln w+3}}{w}=e^{3}$$

Final result: $$f_{W}\left(w\right)=\begin{cases} e^{3} & \text{if }w\in\left(0,e^{-3}\right)\\ 0 & \text{otherwise} \end{cases}$$

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