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 $$ $$Pr(X\ge a)\le \frac{\mathrm{E}(X)}{a}$$

Proof

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

$$\begin{array}{rl}\mathrm{E}(X)& ={\int }_{\omega \subseteq \mathrm{\Omega }}X(\omega )Pr(\omega )\\ & ={\int }_{\omega \subseteq A}X(\omega )Pr(\omega )+{\int }_{\omega \subseteq \mathrm{\Omega }-A}X(\omega )Pr(\omega )\\ & \ge {\int }_{\omega \subseteq A}aPr(\omega )+{\int }_{\omega \subseteq \mathrm{\Omega }-A}0Pr(\omega )\\ & =Pr(A)\end{array}$$

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