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.

Answer 1

Assuming you're talking about the $n$th 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 $n$th moment for the standard uniform distribution is then

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