Markov's bound

Dependencies:

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

Let $X$ be a non-negative random variable and let $a > 0$. Then $\newcommand{\E}{\operatorname{E}}$ \[ \Pr(X \ge a) \le \frac{\E(X)}{a} \]

Proof

Define the event $A$ as $A = \{\omega \in \Omega: X(\omega) \ge a\}$. Then

\begin{align} \E(X) &= \int_{\omega \subseteq \Omega} X(\omega)\Pr(\omega) \\ &= \int_{\omega \subseteq A} X(\omega)\Pr(\omega) + \int_{\omega \subseteq \Omega - A} X(\omega)\Pr(\omega) \\ &\ge \int_{\omega \subseteq A} a\Pr(\omega) + \int_{\omega \subseteq \Omega - A} 0\Pr(\omega) \\ &= \Pr(A) \end{align}

Dependency for:

1. Chebyshev's inequality
2. Cantelli's inequality
3. Chernoff bound

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