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

Result R5285 on D1719: Expectation

Expectation of a gaussian random real number

Formulation 0

Let $G \in \text{Gaussian}(\mu, \sigma)$ be a D210: Gaussian random real number.

Then \begin{equation} \mathbb{E} G = \mu \end{equation}

Proofs

Proof 0

Let $G \in \text{Gaussian}(\mu, \sigma)$ be a D210: Gaussian random real number.

By definition, $G = \sigma Z + \mu$ for some $Z \in \text{Gaussian}(0, 1)$. Using result R4593: Expectation of a standard gaussian random real number, we have \begin{equation} \mathbb{E} G = \mathbb{E}(\sigma Z + \mu) = \mu + \sigma \mathbb{E} Z = \mu \end{equation} $\square$