Vysochanskij–Petunin inequality

Vysochanskij–Petunin Inequality is a theorem in probability theory that refines Chebyshev's inequality by providing a tighter bound under the assumption that the distribution of the random variable is unimodal. This inequality is named after the mathematicians V. F. Vysochanskij and Y. I. Petunin who introduced it in 1976. The Vysochanskij–Petunin inequality states that for any unimodal random variable X with mean μ and finite variance σ^2, and for any k > √2, the probability that X deviates from its mean by more than k standard deviations is less than or equal to 1/(4k^2). Mathematically, it is expressed as:

P(|X - μ| ≥ kσ) ≤ 1/(4k^2)

Background

The Central Limit Theorem and Law of Large Numbers are foundational in understanding the behavior of random variables and their distributions. Chebyshev's inequality provides a non-zero lower bound on the probability that a random variable differs from its mean by more than a certain number of standard deviations, applicable to any distribution with a finite mean and variance. The Vysochanskij–Petunin inequality builds on this by assuming the distribution is unimodal, allowing for a more precise estimation.

Application

This inequality is particularly useful in statistics for bounding probabilities without requiring a detailed knowledge of the distribution's shape, beyond its unimodality. It finds applications in various fields such as quality control, risk management, and econometrics, where it helps in making more accurate predictions and estimations.

Comparison with Chebyshev's Inequality

While Chebyshev's inequality is applicable to any distribution with a finite mean and variance, the Vysochanskij–Petunin inequality requires the additional assumption of unimodality. However, this additional assumption allows for a tighter bound, making the Vysochanskij–Petunin inequality a more powerful tool in certain situations.

Limitations

The main limitation of the Vysochanskij–Petunin inequality is its requirement that the distribution be unimodal. This condition can be restrictive, as many practical distributions may not satisfy this criterion, or it may be difficult to verify unimodality in practice.

See Also

• Probability Theory
• Chebyshev's Inequality
• Central Limit Theorem
• Law of Large Numbers