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The **Lineo-normal distribution** is a member of the modified normal distributions constructed by Romanowski [1].

He states

“The lineo-normal distributions seem to be strikingly well confirmed by types of observations and measurements”.

M. Romanowski

It is a special case of the modified normal cumulative distribution function (CDF) [2]

with *a* = 1.

The above formula is a mixture distribution obtained by giving a power function distribution with density (*a* + 1)*t ^{a}*, 0 ≤

*t*≤ 1 and

*a*≥ 1 to the standard normal distribution

*N*(0,

*σ*)/

^{2}*σ*.

^{2}When *a* tends to infinity, the curve becomes a normal distribution. When *a* = 3 the curve is approximately normal. The ratio between the peak of the modified normal curve and the peak of the corresponding normal curve—with equal variance—is 1.09 for the lineo-normal distribution [3].

## Properties of the Lineo-Normal Distribution

- Symmetric about 0.
- Variance: σ
^{2}(*a*+ 1)/(*a*+ 2) = σ^{2}(1 + 1)/(1 + 2) = 2σ^{2}/3. - Kurtosis: 3(
*a*+ 2)^{2}/{(*a*+ 1)(*a*+ 3)} = 3(3 + 2)^{2}/{(1 + 1)(1 + 3)} = 3(5)^{2}/{(2)(4)} = 75/6 = 12.5.

Related distributions:

- Equi-Normal Distribution (
*a*= 1). - Radico-Normal Distribution (
*a*= ½). - Quadri-Normal Distribution (
*a*= 2).

## References

[1] M. ROMANOWSKI: On the normal law of errors. Bull. Géodésique 73, 95 (1964).

[2] Kotz, S., and Johnson, N. L. (1985). Modified normal distributions, Encyclopedia of Statistical Sciences, 5, S. Kotz, N. L. Johnson, and C. B. Read (editors), 590-591, New York: Wiley.

[3] Maarek, A. & Konecny, G. Modulated Normal Distribution. Online: https://www.asprs.org/wp-content/uploads/pers/1973journal/sep/1973_sep_959-965.pdf