## Pearson System

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.
 (3)
giving

 (4)

 (5)

also,
 (6)

so
 (7)

For ,
 (8)

so
 (9)

For ,
 (10)

so
 (11)

Now let . Then
 (12) (13) (14)

Hence , and so
 (15)

For ,
 (16)

For ,
 (17)

So the Skewness and Kurtosis are
 (18) (19)

So the parameters , , and can be written
 (20) (21) (22)

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.