X ≤ Y ⟹ E(X) ≤ E(Y)

Dependencies:

1. Random variable
2. Expected value of a random variable
3. /measure-theory/lebesgue-integral

$\newcommand{\E}{\operatorname{E}}$ Let $X$ and $Y$ be real-valued random variables over the probability space $(\Omega, \mathcal{F}, \Pr)$. If $X(\omega) \le Y(\omega)$ for each $\omega \in \Omega$, then $\E(X) \le \E(Y)$.

Proof

\[ \E(X) = \int_{\omega \subseteq \Omega} X(\omega)\Pr(\omega) \le \int_{\omega \subseteq \Omega} Y(\omega)\Pr(\omega) = \E(Y) \]

Dependency for:

1. |mean - median| ≤ stddev

Info:

• Depth: 7
• Number of transitive dependencies: 18

Transitive dependencies:

1. /analysis/topological-space
2. /sets-and-relations/countable-set
3. /sets-and-relations/de-morgan-laws
4. /measure-theory/lebesgue-integral
5. σ-algebra
6. Generated σ-algebra
7. Borel algebra
8. Measurable function
9. Generators of the real Borel algebra (incomplete)
10. Measure
11. σ-algebra is closed under countable intersections
12. Group
13. Ring
14. Field
15. Vector Space
16. Probability
17. Random variable
18. Expected value of a random variable