nLab expectation value

Idea

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.

Definition

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

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

In terms of probability monads

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.

Related concepts

• random variable
• correlation function
• moment
• vacuum expectation value
• Feynman diagram
• probability monad
• convex mixture
• mean

References

• Peter Whittle, Probability via expectation, Springer 1992