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Math Help - Stats X uniform over (0,1) calculate E(X^n) and Var(X^n)

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    Stats X uniform over (0,1) calculate E(X^n) and Var(X^n)

    If X is Uniform over (0,1), calculate E(X^n) and Var(X^n).


    Please help
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    Quote Originally Posted by primo iv View Post
    If X is Uniform over (0,1), calculate E(X^n) and Var(X^n).


    Please help
    Definition: E[g(X)] = \int_{-\infty}^{+\infty} g(x) \, f(x) \, dx where f(x) is the pdf of X.

    Therefore E[X^n] = \int_0^1 x^n \, (1) \, dx = \, ....


    Let Y = X^n.

    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.


    --------------------------------------------------------------------------------------

    As an alternative to the above calculations, you could calculate the pdf of Y = X^n.

    It's not hard to get f(y) = \frac{1}{n} y^{\frac{1}{n} - 1} for 0 \leq y \leq 1 and zero otherwise.

    Use it to calculate E(Y) and Var(Y).
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