The probability density function (PDF) of a continuous distribution is defined as the derivative of the (cumulative) distribution function ,
(1)
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(2)
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(3)
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so
(4)
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(5)
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A probability function satisfies
(6)
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and is constrained by the normalization condition,
(7)
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(8)
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Special cases are
(9)
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(10)
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(11)
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(12)
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(13)
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To find the probability function in a set of transformed variables, find the Jacobian. For example, If , then
(14)
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so
(15)
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Similarly, if and , then
(16)
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Given probability functions , , ..., , the sum distribution has probability function
(17)
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where is a delta function. Similarly, the probability function for the distribution of is given by
(18)
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The difference distribution has probability function
(19)
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and the ratio distribution has probability function
(20)
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Given the moments of a distribution (, , and the gamma statistics ), the asymptotic probability function is given by
(21)
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where
(22)
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is the normal distribution, and
(23)
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for (with cumulants and the standard deviation; Abramowitz and Stegun 1972, p. 935).