Vysochanskiï-Petunin inequality
Encyclopedia
In probability theory
, the Vysochanskij–Petunin inequality gives a lower bound for the probability
that a random variable
with finite variance
lies within a certain number of standard deviation
s of the variable's mean
, or equivalently an upper bound for the probability that it lies further away. The sole restriction on the distribution
is that it be unimodal and have finite variance
. (This implies that it is a continuous probability distribution except at the mode
, which may have a non-zero probability.)
The theorem applies even to heavily skewed distributions and puts bounds on how much of the data is, or is not, "in the middle."
Theorem. Let X be a random variable with unimodal distribution, mean μ and finite, non-zero variance σ2. Then, for any λ > √(8/3) = 1.63299...
Furthermore, the limit is attained (that is, the probability is equal to 4/(9 λ2)) for a random variable having a probability 1 − 4/(3 λ2) of being exactly equal to the mean, and which, when it is not equal to the mean, is distributed uniformly in an interval centred on the mean. When λ is less than √(8/3), there exist unsymmetric distributions for which the 4/(9 λ2) limit is exceeded.
The theorem refines Chebyshev's inequality
by including the factor of 4/9, made possible by the condition that the distribution be unimodal.
It is common, in the construction of control chart
s and other statistical heuristics, to set λ = 3, corresponding to an upper probability bound of 4/81 = 0.04938, and to construct 3-sigma limits to bound nearly all (i.e. 95%) of the values of a process output. Without unimodality Chebyshev's inequality would give a looser bound of 1/9 = 0.11111.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, the Vysochanskij–Petunin inequality gives a lower bound for the probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
that a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
with finite variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
lies within a certain number of standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
s of the variable's mean
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
, or equivalently an upper bound for the probability that it lies further away. The sole restriction on the distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
is that it be unimodal and have finite variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
. (This implies that it is a continuous probability distribution except at the mode
Mode (statistics)
In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....
, which may have a non-zero probability.)
The theorem applies even to heavily skewed distributions and puts bounds on how much of the data is, or is not, "in the middle."
Theorem. Let X be a random variable with unimodal distribution, mean μ and finite, non-zero variance σ2. Then, for any λ > √(8/3) = 1.63299...
Furthermore, the limit is attained (that is, the probability is equal to 4/(9 λ2)) for a random variable having a probability 1 − 4/(3 λ2) of being exactly equal to the mean, and which, when it is not equal to the mean, is distributed uniformly in an interval centred on the mean. When λ is less than √(8/3), there exist unsymmetric distributions for which the 4/(9 λ2) limit is exceeded.
The theorem refines Chebyshev's inequality
Chebyshev's inequality
In probability theory, Chebyshev’s inequality guarantees that in any data sample or probability distribution,"nearly all" values are close to the mean — the precise statement being that no more than 1/k2 of the distribution’s values can be more than k standard deviations away from the mean...
by including the factor of 4/9, made possible by the condition that the distribution be unimodal.
It is common, in the construction of control chart
Control chart
Control charts, also known as Shewhart charts or process-behaviour charts, in statistical process control are tools used to determine whether or not a manufacturing or business process is in a state of statistical control.- Overview :...
s and other statistical heuristics, to set λ = 3, corresponding to an upper probability bound of 4/81 = 0.04938, and to construct 3-sigma limits to bound nearly all (i.e. 95%) of the values of a process output. Without unimodality Chebyshev's inequality would give a looser bound of 1/9 = 0.11111.
See also
- Gauss's inequalityGauss's inequalityIn probability theory, Gauss's inequality gives an upper bound on the probability that a unimodal random variable lies more than any given distance from its mode....
, a similar result for the distance from the mode rather than the mean