Chebyshev's inequality

Chebyshev's Inequality is a fundamental theorem in probability theory and statistics, named after the Russian mathematician Pafnuty Chebyshev. It provides a universal bound on the probability that the outcome of a random variable deviates from its mean.

Etymology

The theorem is named after Pafnuty Chebyshev, a prominent 19th-century Russian mathematician who made significant contributions to number theory, probability theory, and mechanics.

Statement of the Theorem

In the field of probability theory, Chebyshev's Inequality states that for a wide class of probability distributions, no more than a certain amount of values can be more than a certain distance from the mean. Specifically, for any random variable with a finite expected value μ and finite non-zero variance σ², the inequality is defined as:

Pr(|X - μ| ≥ kσ) ≤ 1/k²

where:

• Pr denotes the probability
• X is a random variable
• μ is the expected value (mean) of X
• σ is the standard deviation of X
• k is a positive real number

Applications

Chebyshev's Inequality has wide applications in various fields including statistics, economics, computer science, finance, and physics. It is often used to prove the Weak Law of Large Numbers. It is also used in the proof of the Central Limit Theorem.

Related Terms

• Probability Theory
• Random Variable
• Expected Value
• Standard Deviation
• Variance
• Weak Law of Large Numbers
• Central Limit Theorem

See Also

• Markov's Inequality
• Tchebysheff's inequality
• Kolmogorov's inequality

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