nLab
expectation value

Contents

Contents

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,μ)(X, \mu) a measure space of finite total measure Xμ\int_X \mu and for ff an measurable function on XX, a random variable, then its expectation value is

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

In terms of the probability measure μ P1 Xμμ\mu_P \coloneqq \frac{1}{\int_X \mu} \mu this is simply the integral

f= Xfμ P. \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.

References

Last revised on January 22, 2020 at 18:04:46. See the history of this page for a list of all contributions to it.