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Answer by WhoKnowsWho for Understanding proof of $\mathbb{E}[X^n] =...

First a few things about CDF. Let $X$ be any random variable and $F$ be the CDF of $X.$ What we know about $F$ is that $F$ is increasing, right-continuous, and $F(-\infty)=0=1-F(\infty).$ Since $F$ is...

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Understanding proof of $\mathbb{E}[X^n] = \int_{0}^\infty n x^{n-1} (1 -...

For a nonnegative random variable $X$ with CDF $F$ and $n \geq 1$,$\begin{align*}\mathbb{E}[X^n] &= \int_{0}^\infty x^n d F(x)\\&= \int_{0}^\infty \left ( \int_{0}^x n y^{n-1} dy \right ) d...

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