1. Distribution functions

Hi people, I am new here and need a help !

Actually this question is taken from home work in the "statistical inference" course, but it's the second week of the course, so the question is still about probability, so I hope it's ok in this forum...

Let X be a random variable with a probability function f:

f(x;alpha)=alpha*e^x*e^(-alpha*e^x)

( yes, it's e^(e^x).... )

A. find the distribution of the random variable Y=exp(X).
B. Let X1,...Xn be a random sample from this distribution.
calculate the distribution of the random variable:
2alpha*sigma(exp(xi))

I failed in both section A and B....
I have passed both "introduction to probability" and "distribution theory", so it's not an easy question, I think.
I hope anyone here can help me, I am trying to solve it for 2 days now !

It's great to find this forum, I will do my best to help other people as well.

is anyone here know a good website with question and answers about statistical inference ? ( advanced inference, what's called statistical theory )

Thanks a lot to all of you that try to help !!

2. I forgot to tell you what I was trying to do !

well, I tried to find F(x) from f(x), so I can use a transformation:

F(Y)=P(Y<y)=P(E^X<y)=P(x<ln y)=Fx(ln y)

the thing is, that I can't find F from f, the integral is too hard.
so I left it and tried to use moment generating functions, there is a law saying that if Mx(t)=My(t) then X and Y have the same distribution. this idea failed too, maybe it's the correct way ( since it's a statistics course and not probability it make sense...) but the maths failed me once again, the integral is too hard and I couldn't solve it.

I am desperate for ideas...

3. Hi WeeG,

Welcome to the forum. The f(x) looks intimidating. It may be close to some known distribution, possibly related to survival analysis. I don't think you can integrate f(x) as is.

4. I managed to solve this one eventually.
the solution is via transformation. since f(x) is known, you can actually "skip" the step of finding F(x).