Law of the unconscious statistician

In probability theory and statistics, the law of the unconscious statistician, or LOTUS, 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 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 its probability mass function ''ƒ_X'' (but not ''ƒ''_''g''(''X'')), then the expected value of ''g''(''X'') is

: \operatorname (X)= \sum_x g(x) f_X(x), \,

where the sum is over all possible values ''x'' of ''X''. If it is a continuous distribution and one knows its probability density function ''ƒ''_''X'' (but not ''ƒ''_''g''(''X'')), then the expected value of ''g''(''X'') is

: \operatorname (X)= \int_^\infty g(x) f_X(x) \, \mathrmx

If one knows the cumulative probability distribution function ''F''_''X'' (but not ''F''_''g''(''X'')), then the expected value of ''g''(''X'') is given by a Riemann–Stieltjes integral

: \operatorname (X)= \int_^\infty g(x) \, \mathrmF_X(x)

(again assuming ''X'' is real-valued).

Etymology

This proposition is known as the law of the unconscious statistician because of a purported tendency to use the identity without realizing that it must be treated as the result of a rigorously proved theorem, not merely a definition.

Joint distributions

A similar property holds for joint distributions. For discrete random variables ''X'' and ''Y'', a function of two variables ''g'', and joint probability mass function ''f''(''x'', ''y''):

: \operatorname (X, Y)= \sum_y \sum_x g(x, y) f(x, y)

In the absolutely continuous case, with ''f''(''x'', ''y'') being the joint probability density function,

: \operatorname (X, Y)= \int_^\infty \int_^\infty g(x, y) f(x, y) \, \mathrmx \, \mathrmy

Proof

This law is not a trivial result of definitions as it might at first appear, but rather must be proved.

Continuous case

For a continuous random variable ''X'', let ''Y'' = ''g''(''X''), and suppose that ''g'' is differentiable and that its inverse ''g''⁻¹ is monotonic. By the formula for inverse functions and differentiation,

::$\frac(g^(y)) = \frac$

Because ''x'' = ''g''⁻¹(''y''),

::$dx = \fracdy$

So that by a change of variables,

::$\int_^\infty g(x)f_X(x) \, dx = \int_^\infty yf_X(g^(y))\frac \, dy$

Now, notice that because the cumulative distribution function $F_Y(y) = P(Y \leq y)$, substituting in the value of ''g''(''X''), taking the inverse of both sides, and rearranging yields $F_Y(y) = F_X(g^(y))$. Then, by the chain rule,

::$f_Y(y) = f_X(g^(y))\frac$

Combining these expressions, we find

::$\int_^\infty g(x)f_X(x) \, dx = \int_^\infty yf_Y(y) \, dy$

By the definition of expected value,

::\operatorname E (X)= \int_^\infty g(x)f_X(x) \, dx

Discrete case

Let $Y = g(X)$. Then begin with the definition of expected value.

:\operatorname E = \sum_y yf_(y)

:\operatorname E (X)= \sum_y yP(g(X) = y)

:\operatorname E (X)= \sum_y y \sum_ f_X(x)

Rewriting "for all $y$, for all $x$ such that $g(x) = y$" as "for all $x$" since $g^$ is monotonic,

:\operatorname E (X)= \sum_x g(x)f_X(x)

From measure theory

A technically complete derivation of the result is available using arguments in measure theory, in which the probability space of a transformed random variable ''g(X)'' is related to that of the original random variable ''X''. The steps here involve defining a pushforward measure for the transformed space, and the result is then an example of a change of variables formula.

: $\int_\Omega g \circ X \, \mathrmP = \int_ g \, \mathrm(X_* P)$

We say $X:(\Omega, \Sigma, P)\to (\Omega_, \Sigma_)$ has a density if the pushforward measure $\mathrm(X_* P)$ is absolutely continuous with respect to the Lebesgue measure $\mu$. In that case,

: $\mathrm(X_*P) = f \, \mathrm \mu,$

where $f : \to \mathbb$ is the density (see Radon-Nikodym derivative). So the above can be rewritten as the more familiar

: \operatorname (X)= \int_ g \circ X \, \mathrmP = \int_ g(x) f(x) \, \mathrmx .

References

{{Reflist

Theory of probability distributions
Statistical laws