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# Pearson System

A system of equation types obtained by generalizing the differential equation for the normal distribution

 (1)

which has solution

 (2)

to

 (3)

which has solution

 (4)

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)

giving

 (6)
 (7)

Now define the raw th moment by

 (8)

so combining (7) with (8) gives

 (9)

For ,

 (10)

so

 (11)

and for ,

 (12)

so

 (13)

Combining (11), (13), and the definitions

 (14) (15)

obtained by letting and solving simultaneously gives and . Writing

 (16)

then allows the general recurrence to be written

 (17)

For the special cases and , this gives

 (18)
 (19)

so the skewness and kurtosis excess are

 (20) (21)

The parameters , , and can therefore be written

 (22) (23) (24)

where

 (25)

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## 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.

Pearson System

## Cite this as:

Weisstein, Eric W. "Pearson System." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PearsonSystem.html