If X is Uniform over (0,1), calculate E(X^n) and Var(X^n).
Please help
Definition: $\displaystyle E[g(X)] = \int_{-\infty}^{+\infty} g(x) \, f(x) \, dx$ where f(x) is the pdf of X.
Therefore $\displaystyle E[X^n] = \int_0^1 x^n \, (1) \, dx = \, ....$
Let $\displaystyle Y = X^n$.
$\displaystyle Var[Y] = E\left[Y^2\right] - (E[Y])^2 = E\left[(X^n)^2\right] - (E\left[X^n\right])^2 = E\left[X^{2n}\right] - \left(E\left[X^n\right]\right)^2$.
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As an alternative to the above calculations, you could calculate the pdf of Y = X^n.
It's not hard to get $\displaystyle f(y) = \frac{1}{n} y^{\frac{1}{n} - 1}$ for $\displaystyle 0 \leq y \leq 1$ and zero otherwise.
Use it to calculate E(Y) and Var(Y).