The title is self-contained, I'm trying to calculate the PDF of $1000e^{Z\sqrt{5}}$ where $Z$~$N(0,1)$, and the goal is to get the expectation.
The only source I found online that demonstrates how to derive the PDF of a function of a random variable is page 16 of this link: http://www.maths.qmul.ac.uk/~pettit/MTH5122/chapter2.pdf
Letting $Y = 1000e^{Z\sqrt{5}}$ and following the calculation described in that link (a bit too tedious to type here in its entirety) I get:
$$\frac{1}{\sqrt{2 \pi}}\frac{1}{y \sqrt{5}} e^{-\frac{(\frac{1}{\sqrt{5}}ln(\frac{y}{1000}))^2}{2}{}}$$
Is it possible to validate this? If this is incorrect, is there a suggestion or a source you can point me to to help me figure out this density.
Alternatively, maybe there is a way to calculate the expectation of $V(Z)$ through the density of $Z$ directly, but I don't see how that's possible.
1000 is not a problem because it is a constant and thus $E(1000X)=1000E(X)$ and $V(1000X)=1000^2V(X)$
It is very easy to prove that
$$Y=Z\sqrt{5}\sim N(0;5)$$
It is well known that $e^Y$ has a lognormal distribution thus the problem is solved. You can calculate whatever you want given that the random law is known
In the enclosed link (at the paragraph Characteristic function and moment generating function) there is a very easy and quick proof to get all the simple moments of your rv