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Integrable random variable

A random variable is said to be integrable if its expected value exists and it is well-defined.

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Integrability for discrete variables

If X is a discrete random variable having support R_X and probability mass function [eq1], it is integrable if and only if[eq2]

This condition, called absolute summability, guarantees that the expected value[eq3]is well-defined.

Integrability for continuous variables

If X is a continuous random variable having support R_X and probability density function [eq4], it is integrable if and only if[eq5]

This condition, called absolute integrability, guarantees that the expected value[eq6]is well-defined.

Square integrability

A random variable is said to be square integrable if the expected value of its square exists and it is well-defined.

More details

The lectures entitled Expected value and Variance explain these terms in more detail.

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