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