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

If is a discrete random variable having support and probability mass function , it is integrable if and only if

This condition, called absolute summability, guarantees that the expected valueis well-defined.

If is a continuous random variable having support and probability density function , it is integrable if and only if

This condition, called absolute integrability, guarantees that the expected valueis well-defined.

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

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

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