ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5460 on D3363: Estimator bias
Subresult of R3626: Sample variance is an unbiased estimator for the variance of uncorrelated identically distributed random real numbers
Sample variance is an unbiased estimator for the variance of I.I.D. random real numbers
Formulation 0
Let $X_1, X_2, X_3, \ldots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that
(i) $X_1, X_2, X_3, \ldots$ is an D3358: I.I.D. random collection
(ii) \begin{equation} \mathbb{E} |X_1|^2 < \infty \end{equation}
(iii) \begin{equation} \sigma^2 : = \text{Var} X_1 \end{equation}
(iv) $N \in \{ 1, 2, 3, \ldots \}$ is a D5094: Positive integer
(v) \begin{equation} \overline{X}_N : = \frac{1}{N} \sum_{n = 1}^N X_n \end{equation}
Then \begin{equation} \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N |X_n - \overline{X}_N|^2 \right) = \frac{N - 1}{N} \sigma^2 \end{equation}
Proofs
Proof 0
Let $X_1, X_2, X_3, \ldots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that
(i) $X_1, X_2, X_3, \ldots$ is an D3358: I.I.D. random collection
(ii) \begin{equation} \mathbb{E} |X_1|^2 < \infty \end{equation}
(iii) \begin{equation} \sigma^2 : = \text{Var} X_1 \end{equation}
(iv) $N \in \{ 1, 2, 3, \ldots \}$ is a D5094: Positive integer
(v) \begin{equation} \overline{X}_N : = \frac{1}{N} \sum_{n = 1}^N X_n \end{equation}