ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5460 on D3363: Estimator bias
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}