Statistics - Distribution Function Technique

[Addem]

Homework Statement

(From Probability and Statistical Inference, Hogg and Tanis, Eighth Edition, 5.1-5)

The p.d.f. of X is $f(x) = \theta x^{\theta - 1}$ for $0<x<1$ and $0<\theta<\infty$. Let $Y = -2\theta \ln X$. How is $Y$ distributed?

Homework Equations

Um... Fundamental Theorem of Calculus?

The Attempt at a Solution

So I've actually solved this, it's exponential with $\theta = 2$. My question is about the answer given in the back of the book: It cryptically says $M(t) = (1-2t)^{-1}$ for $t<1/2$ which is, to my eye, a useless calculation of the moment-generating function. WTF? Why is this here? Ideas?

Thank you.

 

[cimmerian]

What's "\"?

 

[Addem]

That is a slash. I'm not 100% sure why you ask, but I think maybe it's because you're confused about how to read the 1/2 (one half).

 

[cimmerian]

transformation using the mgf approach?

 

[Ray Vickson]

Homework Statement

(From Probability and Statistical Inference, Hogg and Tanis, Eighth Edition, 5.1-5)

The p.d.f. of X is $f(x) = \theta x^{\theta - 1}$ for $0<x<1$ and $0<\theta<\infty$. Let $Y = -2\theta \ln X$. How is $Y$ distributed?

Homework Equations

Um... Fundamental Theorem of Calculus?

The Attempt at a Solution

So I've actually solved this, it's exponential with $\theta = 2$. My question is about the answer given in the back of the book: It cryptically says $M(t) = (1-2t)^{-1}$ for $t<1/2$ which is, to my eye, a useless calculation of the moment-generating function. WTF? Why is this here? Ideas?

Thank you.

One way do get a density is to get its MGF and hope you obtain a familiar, recognizable form. The form they gave you IS familiar. Of course, you are free to do the problem some other way.

RGV