Hansmann’s Distributions

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:

Hansmann’s Distributions - equation from which they are derived

Which becomes

Hansmann’s Distributions - derived equation becomes another equation

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.

Hansmann’s Distributions - essential correction to probability density functions by Pawula & Rice reveals how these equations can be used when studying varying events equation 1 Hansmann’s Distributions - essential correction to probability density functions by Pawula & Rice reveals how these equations can be used when studying varying events equation 2 Hansmann’s Distributions - essential correction to probability density functions by Pawula & Rice reveals how these equations can be used when studying varying events equation 3

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.


Comments? Need to post a correction? Please Contact Us.