Proof: Monotonicity of the expected value
Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Expected value ▷ Monotonicity
Metadata: ID: P54 | shortcut: mean-mono | author: JoramSoch | date: 2020-02-17, 21:00.
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
\[\label{eq:mean-mono-qed} \mathrm{E}(Y) - \mathrm{E}(X) \geq 0 \; .\]∎
Sources: - Wikipedia (2020): "Expected value"; in: Wikipedia, the free encyclopedia, retrieved on 2020-02-17; URL: https://en.wikipedia.org/wiki/Expected_value#Basic_properties.
Metadata: ID: P54 | shortcut: mean-mono | author: JoramSoch | date: 2020-02-17, 21:00.