| 释义 |
Pearson SystemGeneralizes the differential equation for the Gaussian Distribution
 | (1) |
 | (2) |
 ,  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 isintermediate 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 aMode, 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) |
 | (6) |
 | (7) |
 ,
 | (8) |
 | (9) |
 ,
 | (10) |
 | (11) |
 . Then
Hence  , and  so
 | (15) |
 ,
 | (16) |
 ,
 | (17) |
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.
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