Standard Deviation of transformed R.V.
Suppose you have some random variable x that is distributed according to a pareto distribution f(x) = k mk x-k-1. And then you have a transformation y = c x where c is a constant. I want to find out the standard deviation of the natural log of y. I have been trying for several hours but have had no success. I have tried to do it the standard way E[x^2]-E[x]^2 but am quickly running into a mess that I can't deal with. Does anybody here have any idea how I can approach this problem? I really appreciate the help.
Re: Standard Deviation of transformed R.V.
ive got two suggestions for you:
since y = log c + log x; the sd of y does not depend on the value of c.
so without loss of generality, consider the case c=1.
with C=1, consider the MGF of y=logx
is just the underlying (non central) moment of X. since x follows a standard distribution, you can look this up.
Since you now have a working expression for you can use standard methods to obtain the moments of Y from . (assuming you can differenciate at t=0.)
PS: i haven't actually done this, but i assume it is tractable.
PPS: if this post makes no sense to you, then you probably have not learned "moment generating functions" yet, and youll have to solve your problem the old fashioned way.