Law of the unconscious statistician
Encyclopedia
In probability theory
and statistics
, the law of the unconscious statistician is a theorem used to calculate the expected value
of a function
g(X) of a random variable
X when one knows the probability distribution
of X but one does not explicitly know the distribution of g(X).
The form of the law can depend on the form in which one states the probability distribution of the random variable X. If it is a discrete distribution and one knows the probability mass function
ƒ of X (not of g(X)), then the expected value of g(X) is
where the sum is over all possible values x of X. If it is a continuous distribution and one knows the probability density function
ƒ of X (not of g(X)), then the expected value of g(X) is
(provided the values of X are real numbers as opposed to vectors, complex numbers, etc.).
Regardless of continuity-versus-discreteness and related issues, if one knows the cumulative probability distribution function F of X (not of g(X)), then the expected value of g(X) is given by a Riemann–Stieltjes integral
(again assuming X is real-valued).
However, the result is so well-known that it is usually used without stating a name for it: the name is not extensively used. For justifications of the result for discrete and continuous random variables see expected value
.
is measurable, then
remembers 
and accordingly induces a natural pushforward measure
on the target space,
, given by taking measures of preimages. Informally, the pushforward asks: "What's the probability I would land in 
if I were to be randomly spawned in 
and followed 
into 
?"
It then follows that for any measurable
,
Intuitively, this says that the pushforward measure really gives no new information. When the above is specialized to
, the result is the Law of the Unconscious Statistician.
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...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the law of the unconscious statistician is a theorem used to calculate the expected value
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...
of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
g(X) of 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...
X when one knows the probability 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....
of X but one does not explicitly know the distribution of g(X).
The form of the law can depend on the form in which one states the probability distribution of the random variable X. If it is a discrete distribution and one knows the probability mass function
Probability mass function
In probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...
ƒ of X (not of g(X)), then the expected value of g(X) is
where the sum is over all possible values x of X. If it is a continuous distribution and one knows the probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
ƒ of X (not of g(X)), then the expected value of g(X) is
(provided the values of X are real numbers as opposed to vectors, complex numbers, etc.).
Regardless of continuity-versus-discreteness and related issues, if one knows the cumulative probability distribution function F of X (not of g(X)), then the expected value of g(X) is given by a Riemann–Stieltjes integral
(again assuming X is real-valued).
However, the result is so well-known that it is usually used without stating a name for it: the name is not extensively used. For justifications of the result for discrete and continuous random variables see expected value
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...
.
From the perspective of measure
The Law of the Unconscious Statistician is great motivation for a (slightly) more general principle: ifis measurable, then
remembers and accordingly induces a natural pushforward measurePushforward measure
In measure theory, a pushforward measure is obtained by transferring a measure from one measurable space to another using a measurable function.-Definition:...
on the target space,
, given by taking measures of preimages. Informally, the pushforward asks: "What's the probability I would land in if I were to be randomly spawned in and followed into ?"It then follows that for any measurable
,Intuitively, this says that the pushforward measure really gives no new information. When the above is specialized to
, the result is the Law of the Unconscious Statistician.