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