Distribution functions

[WeeG]

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 !!

[WeeG]

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...

[quark]

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.

[WeeG]

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).
the final answer is X~exp(alpha).

thanks for trying to help ! I'll visit here more often now.
I have a bachelor degree in computer science and I am studing for a master degree in statistics, so during the master I have to complete some extra courses in probability and statistics, from the bachelor degree level ( advanced courses of first degree level ). so in this semester I am studing "statistical models B" and "statistical inference", which is statistical theory, part two of the course "distribution theory".

I'll be happy to try and help other people around here, as much as I can.