## Find $E[(1-X)^k]$ given $E[X^k] = \frac{1}{(k+1)^2}$

Find $$E[(1-X)^k]$$ given $$E[X^k] = \frac{1}{(k+1)^2}$$

The previous part of the question involved finding $$E[X^k]$$ given $$X=AB$$, where $$A$$ and $$B$$ are uniformly distributed over $$[0, 1]$$. However, I don’t think this helps with the matter at hand.

My intuition was to expand $$E[(1-X)^k]$$ as a series, which gave me a nasty sum I couldn’t compress. Is there a more direct approach? Can I avoid using $$A$$ and $$B$$?