In probability theory the *expectation value* of a *random variable* or *observable* is to be thought of as the *mean* value of that variable/observable under the given probabilities.

Taking the concept of expectation value as the primary concept (Whittle 92) leads to *quantum probability theory*.

For $(X, \mu)$ a measure space of finite total measure $\int_X \mu$ and for $f$ an measurable function on $X$, a *random variable*, then its **expectation value** is

$\langle f\rangle
\coloneqq
\frac{\int_X f \cdot \mu}{\int_X \mu}
\,.$

In terms of the probability measure $\mu_P \coloneqq \frac{1}{\int_X \mu} \mu$ this is simply the integral

$\langle f\rangle
=
\int_X f \cdot \mu_P
\,.$

For classical probability (not quantum), spaces equipped with a notion of expectation value can be modeled as algebras over a probability monad. See probability monad - algebras for more.

- Peter Whittle,
*Probability via expectation*, Springer 1992

Last revised on September 8, 2022 at 10:38:52. See the history of this page for a list of all contributions to it.