Expectation

Definition

Continuous case

(1)
\begin{align} \operatorname{E}(X) = \int_\Omega X\, \operatorname{d}P \end{align}

Discrete case

(2)
\begin{align} \operatorname{E}(X) = \sum_i x_i p(x_i) \, \end{align}

Properties

Linearity

The expected value operator (or '''expectation operator''') E is linear in the sense that

(3)
\begin{align} \operatorname{E}(X + c)= \operatorname{E}(X) + c\, \end{align}
(4)
\begin{align} \operatorname{E}(X + Y)= \operatorname{E}(X) + \operatorname{E}(Y)\, \end{align}
(5)
\begin{align} \operatorname{E}(aX)= a \operatorname{E}(X)\, \end{align}

Note that the second result is valid even if ''X'' is not statistically independent of ''Y''.

Combining the results from previous three equations, we can see that

(6)
\begin{align} \operatorname{E}(aX + b)= a \operatorname{E}(X) + b\, \end{align}
(7)
\begin{align} \operatorname{E}(a X + b Y) = a \operatorname{E}(X) + b \operatorname{E}(Y)\, \end{align}
Bibliography
1. wikipedia
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