Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Expected value ▷ Monotonicity

Theorem: The expected value is monotonic, i.e.

$\label{eq:mean-mono} \mathrm{E}(X) \leq \mathrm{E}(Y), \quad \text{if} \quad X \leq Y \; .$

Proof: Let $Z = Y - X$. Due to the linearity of the expected value, we have

$\label{eq:mean-XYZ} \mathrm{E}(Z) = \mathrm{E}(Y-X) = \mathrm{E}(Y) - \mathrm{E}(X) \; .$

With the non-negativity property of the expected value, it also holds that

$\label{eq:mean-Z} Z \geq 0 \quad \Rightarrow \quad \mathrm{E}(Z) \geq 0 \; .$

Together with \eqref{eq:mean-XYZ}, this yields

$\label{eq:mean-mono-qed} \mathrm{E}(Y) - \mathrm{E}(X) \geq 0 \; .$
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Metadata: ID: P54 | shortcut: mean-mono | author: JoramSoch | date: 2020-02-17, 21:00.