Generalizes the differential equation for the Gaussian Distribution
|
(1) |
to
|
(2) |
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.
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|
|
(3) |
giving
|
|
|
(4) |
|
(5) |
also,
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(6) |
so
|
(7) |
For ,
|
(8) |
so
|
(9) |
For ,
|
(10) |
so
|
(11) |
Now let
. Then
Hence , and so
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(15) |
For ,
|
(16) |
For ,
|
(17) |
So the Skewness and Kurtosis are
So the parameters , , and can be written
where
|
(23) |
References
Craig, C. C. ``A New Exposition and Chart for the Pearson System of Frequency Curves.'' Ann. Math. Stat. 7, 16-28, 1936.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 107, 1951.
Pearson, K. ``Second Supplement to a Memoir on Skew Variation.'' Phil. Trans. A 216, 429-457, 1916.
© 1996-9 Eric W. Weisstein
1999-05-26