[[Estimator]]
# Sample moment

Let $\{ X_{j} \}_{j=1}^n$ be identically and [[Independence of random variables|independently distributed]] [[Real random variable|real random variables]].
The **$k$th sample moment** is #m/def/prob 
$$
\begin{align*}
M_{k} = \frac{1}{n} \sum_{j=1}^n (X_{j})^k
\end{align*}
$$
Note that $\Ex[M_{k}] = \mu_{k}$ whence $M_{k}$ is an [[estimator]] of the $k$th [[Statistical moment|moment]] by the [[Law of large numbers]].
The first sample moment, $M_{1} = \overline{X}_{n}$ is the [[𝜇-estimator]].

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