Question: Suppose that \(X_{1}, \ldots X_{n}\) is a random sample from a population with mean \(\mu\) and unknown variance \(\sigma^{2}\). That is \(E\left(X_{i}\right)=\mu\) and \(V\left(X_{i}\right)=\sigma^{2}\) for each \(\mathrm{i}\).

Prove that \(\bar{X}=\sum_{i=1}^{n} X_{i}\) is an unbiased estimator of \(\mu .\) That is, \(E(\bar{X})=\mu\).

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