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The integral of a random variable with respect to the underlying probability measure.
The expectation of an integrable random variable is
where integrability means .
This definition uses the Lebesgue integral on the underlying probability space; expectation is the basic averaging operation underlying variance, covariance, and many limit theorems.
A random variable is a measurable function defined on a probability space , meaning that for every Borel set one has .
A Lebesgue integral on a measure space assigns a value to a measurable function by reducing to the nonnegative case. Define the positive and negative parts
so that and are nonnegative measurable functions. If at least one of or is finite, define
where the integrals on the right are understood via the nonnegative Lebesgue integral. If both and are , the Lebesgue integral of is left undefined.
When is Lebesgue integrable, the integral is a finite real number. Moreover, if and satisfy a.e. equality, then (whenever defined) their Lebesgue integrals agree.
A probability space is a triple where is a set, is a sigma-algebra on , and is a probability measure on .
Elements of are the events whose probabilities are evaluated by , and a random variable is a measurable function defined on .
A variance of a random variable is the quantity
defined when (so in particular the expectation is finite).
A covariance is the quantity associated to two random variables and with finite second moments.