# Calculating the density function of a variable based on a relation [on hold]

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

## put on hold as off-topic by José Carlos Santos, Brahadeesh, amWhy, Shailesh, LeucippusDec 10 at 17:40

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Brahadeesh, amWhy, Shailesh, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.

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