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

Result R4659 on D5074: Empirical probability distribution function

Empirical probability distribution function is an unbiased estimator of the true distribution function

Formulation 0

Let $X_1, \ldots, X_N \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, \ldots, X_N$ is an D3357: Identically distributed random collection

Let $x \in \mathbb{R}$ be a D993: Real number.

Then \begin{equation} \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \leq x \}} \right) = \mathbb{P}(X_1 \leq x) \end{equation}

Formulation 1

Let $X_1, \ldots, X_N \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, \ldots, X_N$ is an D3357: Identically distributed random collection
(ii)	$F : \mathbb{R} \to [0, 1]$ is a D205: Probability distribution function for $X_1$

Let $x \in \mathbb{R}$ be a D993: Real number.

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

Proofs

Proof 0

Let $X_1, \ldots, X_N \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, \ldots, X_N$ is an D3357: Identically distributed random collection

Let $x \in \mathbb{R}$ be a D993: Real number.

We have \begin{equation} \begin{split} \mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \leq x \}} \right) & = \frac{1}{N} \sum_{n = 1}^N \mathbb{E} (I_{\{ X_n \leq x \}}) \\ & = \frac{1}{N} N \mathbb{E} (I_{\{ X_1 \leq x \}}) = \mathbb{P}(X_1 \leq x) \end{split} \end{equation} $\square$