ThmDex – An index of mathematical definitions, results, and conjectures.
Real-linearity of complex expectation
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $Z, W : \Omega \to \mathbb{C}$ are each a D4877: Random complex number on $P$
(ii) \begin{equation} \mathbb{E} |Z|, \mathbb{E} |W| < \infty \end{equation}
(iii) $\alpha, \beta \in \mathbb{R}$ are each a D993: Real number
Then \begin{equation} \mathbb{E}(\alpha Z + \beta W) = \alpha \mathbb{E} Z + \beta \mathbb{E} W \end{equation}
Formulation 1
Let $Z, W \in \text{Random}(\Omega \to \mathbb{C})$ each be a D4877: Random complex number such that
(i) \begin{equation} \mathbb{E} |Z|, \mathbb{E} |W| < \infty \end{equation}
Let $\alpha, \beta \in \mathbb{R}$ each be a D993: Real number.
Then \begin{equation} \mathbb{E}(\alpha Z + \beta W) = \alpha \mathbb{E} Z + \beta \mathbb{E} W \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $Z, W : \Omega \to \mathbb{C}$ are each a D4877: Random complex number on $P$
(ii) \begin{equation} \mathbb{E} |Z|, \mathbb{E} |W| < \infty \end{equation}
(iii) $\alpha, \beta \in \mathbb{R}$ are each a D993: Real number
This result is a particular case of R1817: Complex-linearity of complex expectation. $\square$