- #1
blalien
- 32
- 0
Homework Statement
The problem is to find the moments [itex]E(X^k)[/itex] of [itex]f_x(x) = (\theta+1)(1-x)^\theta[/itex], [itex]0 < x < 1[/itex], [itex]\theta > -1[/itex]
Homework Equations
[itex]E(X^k)=\int_0^1 x^k (\theta+1)(1-x)^\theta dx[/itex]
According to Mathematica, the solution is [itex]\frac{\Gamma(1+k)\Gamma(2+\theta)}{\Gamma(2+k+ \theta )}[/itex]. I have no idea how to solve this integral by hand, however.
The Attempt at a Solution
If we let [itex]W = -\log(1-x)[/itex], then the distribution for [itex]W[/itex] is [itex]f_w(w) = (\theta+1)e^{-(\theta+1)w}[/itex], which is just the exponential distribution. The moments are [itex]E(W^k) = \frac{\Gamma(k+1)}{(1+\theta)^k}[/itex]. The question is, if we know the relation between x and w and we know the moments for w, is it possible to find the moments for x? Or is there another way to solve this integral?
Thanks in advance!