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

Result R4973 on D5210: Random real column matrix
Subresult of R4972: Bias-variance partition of mean square error for a random real column matrix

Bias-variance partition of mean square error for random basic real number

Formulation 0

Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	\begin{equation} \mathbb{E} \|X\|^2 < \infty \end{equation}

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

Then \begin{equation} \mathbb{E} |X - a|^2 = \text{Var} X - |\mathbb{E} X - a|^2 \end{equation}

Formulation 1

Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	\begin{equation} \mathbb{E} \|X\|^2 < \infty \end{equation}

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

Then \begin{equation} \mathbb{E} |X - a|^2 = \mathbb{E} |X - \mathbb{E} X|^2 - |\mathbb{E} X - a|^2 \end{equation}

Proofs

Proof 0

Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	\begin{equation} \mathbb{E} \|X\|^2 < \infty \end{equation}

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