Probability Distribution: List of Statistical Distributions >
Hansmann’s distributions [1] are an intriguing phenomenon created through a generalized Pearson differential equation[2], creating mathematically-equal symmetry around zero..
They are derived from:
Which becomes
Johnson et al [3] pointed out an essential correction to probability density functions by Pawula & Rice [4]. This valuable insight further reveals how these equations can be used when studying varying events.
Where
- K= ½ {c2(b2 – a2)}-1
- K, K1, K2 = normalizing constants.
K1 and K2 must satisfy
- ∫P(x)dx = 1
- ∫x2p(x)= σ2x.
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.