Chebyshev's inequality

Dependencies:

1. Expected value of a random variable
2. Variance of a random variable
3. Markov's bound

$\newcommand{\E}{\operatorname{E}}$ $\newcommand{\Var}{\operatorname{Var}}$ Let $X$ be a real-valued random variable and let $a > 0$. Then \[ \Pr(|X - \E(X)| \ge a) \le \frac{\Var(X)}{a^2} \]

Proof

\begin{align} \Pr(|X - \E(X)| \ge a) &= \Pr((X - \E(X))^2 \ge a^2) \\ &\le \frac{\E((X - \E(X))^2)}{a^2} \tag{by Markov's bound} \\ &= \frac{\Var(X)}{a^2} \end{align}

Dependency for: None

Info:

• Depth: 9
• Number of transitive dependencies: 22

Transitive dependencies:

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