Probability: Determining the distribution and range of a random variable

[ruby_duby]

Homework Statement

The RV X has parameter p>0 and distribution:

f_X(x) = pxe^-px for x \geq 0 and is 0 otherwise

(The subscript X is a capital letter, as is the X mentioned below in the e^4X)

If we are to consider the RV D= e^4X, determine the range and distribution f_D(d)

Homework Equations

The Attempt at a Solution

I found this question as an exercise in a textbook, but do not know how to answer it!
Does anyone have any suggestions that I can try out please?

 

[LCKurtz]

Start by figuring out what p must be for f_X to be a density function.

If D = e^4X, one way to get at its density function is to use the cumulative distribution:

P(D ≤ d) = P(e^4X ≤ d) = P(4X ≤ ln(d)) = P( X ≤ (1/4)ln(d) )

Presumably you can calculate that, which will give you the cumulative distribution function for D. Differentiate to get f_D(d).