A system of equation types obtained by generalizing the differential equation for the normal distribution
(1)
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which has solution
(2)
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to
(3)
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which has solution
(4)
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Let , be the roots of . Then the possible types of curves are
0. , . E.g., normal distribution.
I. , . E.g., beta distribution.
II. , , where .
III. , , where . E.g., gamma distribution. This case is intermediate to cases I and VI.
IV. , .
V. , where . Intermediate to cases IV and VI.
VI. , where is the larger root. E.g., beta prime distribution.
VII. , , . E.g., Student's t-distribution.
Classes IX-XII are discussed in Pearson (1916). See also Craig (in Kenney and Keeping 1951).
If a Pearson curve possesses a mode, it will be at . Let at and , where these may be or . If also vanishes at , , then the th moment and th moments exist.
(5)
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giving
(6)
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(7)
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Now define the raw th moment by
(8)
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so combining (7) with (8) gives
(9)
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For ,
(10)
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so
(11)
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and for ,
(12)
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so
(13)
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Combining (11), (13), and the definitions
(14)
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(15)
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obtained by letting and solving simultaneously gives and . Writing
(16)
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then allows the general recurrence to be written
(17)
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For the special cases and , this gives
(18)
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(19)
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so the skewness and kurtosis excess are
(20)
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(21)
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The parameters , , and can therefore be written
(22)
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(23)
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(24)
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where
(25)
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