Mean-preserving Spread
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probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
and
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a mean-preserving spread (MPS) is a change from one
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
A to another probability distribution B, where B is formed by spreading out one or more portions of A's
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
or
probability mass function In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
while leaving the mean (the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
) unchanged. As such, the concept of mean-preserving spreads provides a
stochastic ordering In probability theory and statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another. These are usually partial orders, so that one random variable A may be neither stochastically greater than, less tha ...
of equal-mean gambles (probability distributions) according to their degree of
risk In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environme ...
; this ordering is
partial Partial may refer to: Mathematics * Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial ...
, meaning that of two equal-mean gambles, it is not necessarily true that either is a mean-preserving spread of the other. Distribution A is said to be a mean-preserving contraction of B if B is a mean-preserving spread of A. Ranking gambles by mean-preserving spreads is a special case of ranking gambles by second-order
stochastic dominance Stochastic dominance is a partial order between random variables. It is a form of stochastic ordering. The concept arises in decision theory and decision analysis in situations where one gamble (a probability distribution over possible outcomes, ...
– namely, the special case of equal means: If B is a mean-preserving spread of A, then A is second-order stochastically dominant over B; and the
converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical c ...
holds if A and B have equal means. If B is a mean-preserving spread of A, then B has a higher variance than A and the expected values of A and B are identical; but the converse is not in general true, because the variance is a complete ordering while ordering by mean-preserving spreads is only partial.


Example

This example shows that to have a mean-preserving spread does not require that all or most of the probability mass move away from the mean. Let A have equal probabilities 1/100 on each outcome x_ , with x_=198 for i=1,\dots, 50 and x_=202 for i=51,\dots,100; and let B have equal probabilities 1/100 on each outcome x_, with x_=100, x_=200 for i=2,\dots,99, and x_=300. Here B has been constructed from A by moving one chunk of 1% probability from 198 to 100 and moving 49 probability chunks from 198 to 200, and then moving one probability chunk from 202 to 300 and moving 49 probability chunks from 202 to 200. This sequence of two mean-preserving spreads is itself a mean-preserving spread, despite the fact that 98% of the probability mass has moved to the mean (200).


Mathematical definitions

Let x_A and x_B be the random variables associated with gambles A and B. Then B is a mean-preserving spread of A if and only if x_B \overset (x_A + z) for some random variable z having E(z\mid x_A)=0 for all values of x_A. Here \overset means " is equal in distribution to" (that is, "has the same distribution as"). Mean-preserving spreads can also be defined in terms of the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
s F_A and F_B of A and B. If A and B have equal means, B is a mean-preserving spread of A if and only if the area under F_A from minus infinity to x is less than or equal to that under F_B from minus infinity to x for all real numbers x, with strict inequality at some x. Both of these mathematical definitions replicate those of second-order stochastic dominance for the case of equal means.


Relation to expected utility theory

If B is a mean-preserving spread of A then A will be preferred by all
expected utility The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on the ...
maximizers having concave utility. The converse also holds: if A and B have equal means and A is preferred by all expected utility maximizers having concave utility, then B is a mean-preserving spread of A.


See also

*
Stochastic ordering In probability theory and statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another. These are usually partial orders, so that one random variable A may be neither stochastically greater than, less tha ...
*
Risk (statistics) Statistical risk is a quantification of a situation's risk using statistical methods. These methods can be used to estimate a probability distribution for the outcome of a specific variable, or at least one or more key parameters of that distribu ...
*
Scale parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family o ...


References


Further reading

* {{DEFAULTSORT:Mean-Preserving Spread Theory of probability distributions Decision theory