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Hansmann’s distributions [1], obtained from a generalized Pearson differential equation [2], are symmetric about zero.

They are derived from Pearson’s equation

Which becomes

The probability density functions [which Johnson et al [3] notes contain a correction by Pawula & Rice [4]) are:

Where

- K= ½ {c
_{2}(b^{2}– a^{2})}^{-1} - K, K
_{1}, K_{2}are normalizing constants.

K_{1} and K_{2} must satisfy the conditions

- ∫P(x)dx = 1
- ∫x
^{2}p(x)= σ^{2}_{x}.

## References

[1] Hansmann, G. H. (1934). On certain non-normal symmetric frequency distributions, Biometrika, 26, 129-135

[2] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[3] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

[4] Pawula, R. F., and Rice, S. 0. (1989). A note on Hansmann’s 1934 family of distributions, IEEE Transactions on Information Theory, 35, 910-91 1.