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Let $X$ and $Y$ be two independent uniformly distributed random variables on $[0, 1]$. Show that $E[X^k] = \frac{1}{k+1}$ and $E[(XY)^k] = \frac{1}{(k+1)^2}$.

For the first part, I used $M_{X}(t) = \frac{e^t-1}{t} = 1+t\left( \frac{1}{2}\right)+\frac{t^2}{2!}\left( \frac{1}{3}\right)++\frac{t^3}{3!}\left( \frac{1}{4}\right)+...$ and compared this to $M_{X}(t) = E[e^{tX}] = 1+tE[X]+\frac{t^2}{2!}E[X^2]+...$. How valid is it to compare two sums like this? Is there another approach (which is maybe better)?

For the second part, it just got me thinking: is $(XY)^k = X^kY^k$ for $X, Y$ independent only?

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You definitely can compare this way at least if they converge in some neighbor of $0$ - for example, if they are equal, then $n$-th derivatives at $0$ are also equal, and this derivatives are exactly coefficients multiplied by the same constant.

For the second part - $(XY)^k = X^k Y^k$ for any $X$, $Y$ (as multiplication is commutative), but $\mathbb{E}[XY] = \mathbb{E}[X] \cdot \mathbb{E}[Y]$ doesn't hold in general if $X$ and $Y$ are not independent.

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