Let X be a uniform (0, 1) random variable. Compute E[X n ] by using Proposition 2.1, and then check the result by using the definition of ex

Question

Let X be a uniform (0, 1) random variable. Compute E[X n ] by using Proposition 2.1, and then check the result by using the definition of expectation.

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Luna 2 weeks 2021-09-12T04:50:48+00:00 1 Answer 0

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    2021-09-12T04:52:18+00:00

    Assuming you’re talking about the nth moment, E[X^n], we have by definition of expectation of a function of a random variable

    \displaystyle E[X^n]=\int_{-\infty}^\infty x^nf_X(x)\,\mathrm dx

    where f_X is the PDF for X. The nth moment for the standard uniform distribution is then

    \displaystyle\int_0^1 x^n\,\mathrm dx=\frac{x^{n+1}}{n+1}\bigg|_0^1=\boxed{\frac1{n+1}}

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