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

It is a special case of the modified normal cumulative distribution function (CDF) [1]

with a = ½.

Note that when a is infinitely large, the curve is 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 1.16 for the Radico-normal distribution [2].

Properties of the Radico-Normal Distribution

• Symmetric about 0.
• Variance: σ²(a + 1)/(a + 2) = σ²(½ + 1)/( ½ + 2) = σ²(1½ )/( 2½).
• Kurtosis: 3(a + 2)²/{(a + 1)(a + 3)} = 3(½ + 2)²/{(½ + 1)( ½ + 3)} = 3(2½)²/{(1½)( 3½)} ≈ 3.57.

Related distributions:

• Equi-Normal Distribution (a = 1).
• Lineo-Normal Distribution (a = 1).
• Quadri-Normal Distribution (a = 2).

References

[1] M. ROMANOWSKI: Bull. Géodésique 73, 95 (1964).

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