ThmDex – An index of mathematical definitions, results, and conjectures.

Sample mean of identically distributed random real numbers is an unbiased estimator for expectation

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 D3357: Identically distributed random collection
(ii)	\begin{equation} \mathbb{E} \|X_1\| < \infty \end{equation}
(iii)	\begin{equation} \mu : = \mathbb{E} X_1 \end{equation}

Then \begin{equation} \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N X_n \right) = \mu \end{equation}

Formulation 1

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 D3357: Identically distributed random collection
(ii)	\begin{equation} \mathbb{E} \|X_1\| < \infty \end{equation}
(iii)	\begin{equation} \mu : = \mathbb{E} X_1 \end{equation}
(iv)	\begin{equation} \overline{X}_N : = \frac{1}{N} \sum_{n = 1}^N X_n \end{equation}

Then \begin{equation} \mathbb{E} \overline{X}_N = \mu \end{equation}