Given a random variable and a probability density function , if there exists an such that
(1)
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for , where denotes the expectation value of , then is called the moment-generating function.
For a continuous distribution,
(2)
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(3)
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(4)
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where is the th raw moment.
For independent and , the moment-generating function satisfies
(5)
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(6)
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(7)
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(8)
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If is differentiable at zero, then the th moments about the origin are given by
(9)
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(10)
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(11)
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(12)
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The mean and variance are therefore
(13)
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(14)
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(15)
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(16)
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It is also true that
(17)
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where and is the th raw moment.
It is sometimes simpler to work with the logarithm of the moment-generating function, which is also called the cumulant-generating function, and is defined by
(18)
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(19)
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(20)
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But , so
(21)
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(22)
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