Law of the Unconscious Statistician

The law of the unconscious statistician, commonly abbreviated LOTUS (Pishro-Nik 2014, §3.2.3; Blitzstein and Hwang 2019, p. 170; Tsun 2020, theorem 3.2.13), states that the expectation value of a function of a random variable can be computed directly from the statistical distribution of , without first finding the statistical distribution of . In terms of the distribution function ,

(1)

provided the corresponding expectation value exists. For a discrete random variable taking the value with probability , this becomes

(2)

while for a continuous random variable with probability density function , it becomes

(3)

The law is useful because it avoids deriving the statistical distribution of each transformed random variable. For example, if has the uniform distribution on , then

(4)

More generally, choices such as , , and give moments, the variance, and the moment-generating function, respectively. Expected losses, costs, and payoffs can be computed in the same way.

The name jokes about "unconscious" statisticians who use the formula as though it were the definition of , without consciously deriving the statistical distribution of or noting that the formula is a theorem. The name was already in print as the "theorem of the unconscious statistician" by 1965 (Hillier and Lieberman 1965). Ross (1980) used "law of the unconscious statistician" and gave this explanation in a footnote.