Moment-generating function

The moment-generating function of a real random variable or random vector is defined as #m/def/prob

provided it is finite on some interval around 0, otherwise the moment-generating function does not exist.

tip

To make sure a moment-generating function is valid, check that .

If the moment-generating functions of two real random variables match in an arbitrarily small neighbourhood of 0, they must have the same distribution. #m/thm/prob Thus moment-generating function carries all relevant information about the distribution of , and therefore provides an alternative to working with the probability density function and cumulative distribution function directly.

Relation to moments

Taking the Taylor expansion of it follows

the th moment of . #m/thm/prob Hence is a generating function for the moments of .

Properties

for independently distributed

Proof of 1–2

By definition, if it exists

by ^P3, proving ^P1. Similarly

proving ^P2.

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