Chebyshevetiäisyys

Chebyshevetiäisyys, known in English as Chebyshev's inequality, is a fundamental concept in probability theory and statistics. It provides an upper bound on the probability that a random variable will deviate from its expected value by more than a certain amount. Specifically, for any random variable X with a finite expected value μ and a finite non-zero variance σ², Chebyshev's inequality states that for any real number k > 0, the probability that X deviates from its mean by more than k standard deviations is at most 1/k².

In mathematical notation, this is expressed as P(|X - μ| ≥ kσ) ≤ 1/k². This inequality is powerful because it holds true for any probability distribution, regardless of its specific shape. It does not require the distribution to be normal or symmetric.

The implication of Chebyshev's inequality is that even if we have very little information about a random variable, we can still make statements about the likelihood of its values being far from the mean. For example, if k=2, the probability of a random variable deviating from its mean by more than two standard deviations is at most 1/4. If k=3, the probability is at most 1/9.

While Chebyshev's inequality provides a guaranteed bound, it is often a loose bound. For distributions where more information is known, such as the normal distribution, tighter bounds can be derived. Nevertheless, its universality makes it a valuable tool in situations where distributional assumptions cannot be made. It is particularly useful in introductory probability courses and in applied statistics when dealing with limited data or unknown distributions.

---