Pearson System of Distributions

The Pearson system of distributions, named after Karl Pearson (1857 to 1936), includes a wide range of curves including:

• Type I: Beta distribution and (as a limit) Continuous uniform distribution.
• Type III: Chi-squared distribution, Exponential distribution, Gamma distribution.
• Type IV: Cauchy distribution.
• Type VI: Beta prime distribution, F-distribution.
• Type V: Inverse-chi-squared distribution, Inverse-gamma distribution.
• Type VII: Student’s t-distribution (the non-skewed subtype of type IV).

Differential Equations in Pearson System of Distributions

All curves in the system satisfy a differential equation of the form [1]

The exact shape of the distribution depends on the values of parameters a and c.

If –a is not a root of the denominator set to equal zero, i.e.,

then p is finite when x = –a and dp/dx = 0. When p = 0, the slope is zero. If x ≠ –a and p # 0, then dp(x)/dx ≠ 0. However, two conditions must be satisfied:

1. P(x) ≥ 0: The probability of any “x” must be greater than or equal to zero, and
2. Like all probability density functions, the area under the curve must equal 1. In other words, it satisfies the integral

The differential equation

The differential equation tells us that p(x) must tend to zero as x tends to infinity, so must dp/dx.  In some formal solutions, the condition P(x) ≥ 0 is not satisfied, so we must restrict the range of x to values which meet the condition. When x is outside the specified range, we can assign the value p(x) = 0.

Pearson classified a variety of different shapes and types of distributions according to the nature of the roots of the equation

Although the system is well known and widely available in the literature, there isn’t a clear systematic basis to it. However, all the different types correspond to different forms of solution to the above equations. Pearson‘s first papers [2] covered Types I, III, IV, V, and VI. Later [3], he introduced further special cases and subtypes (VII through XII).

For example, if c₁ = c₂=0, the differential equation becomes

Where

And K is a constant chosen to make the area under the curve equal to 1, i.e.,

From here, we can deduce that c₀ must be positive and

This corresponds to a normal distribution with an expected value (i.e., a mean) of -a and standard deviation √c_0. The normal distribution is now viewed as the limit of type I, III, IV, V, or VI.

Toranzos distribution

The Toranzos distribution is an asymmetric, bell-shaped generalization of the Pearson family of distributions.

Toranzos [4] postulated that a random variable’s frequency distribution can be expressed as a product of two functions:

The product can be derived from a differential equation

where

• Q_m+1(x) = a polynomial of degree m + 1
• P_m(x) = a polynomial of degree m.

The differential equation generalizes the Pearson family and can be expressed as a long division of polynomials

References

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

[2] Pearson, Karl (1895). “Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material” (PDF). Philosophical Transactions of the Royal Society. 186: 343–314. Bibcode:1895RSPTA.186..343P. doi:10.1098/rsta.1895.0010. JSTOR 90649.

[3] Pearson, Karl (1901). “Mathematical contributions to the theory of evolution, X: Supplement to a memoir on skew variation”. Philosophical Transactions of the Royal Society A. 197 (287–299): 443–459. Bibcode:1901RSPTA.197..443P. doi:10.1098/rsta.1901.0023. JSTOR 90841.

[4] Toranzos, Fausto. I. (1952). An asymmetric bell-shaped frequency curve. Ann. Math. Statist., 23, 467–469.

[5] Singh, V. & Zhang, L. (2020). Systems of Frequency Distributions for Water and Environmental Engineering. Cambridge University Press.