Pearson Curves

a family of distribution curves—that is, curves y = y(x) representing the dependence on x of the probability density—that satisfy the differential equation

where a, b₀, b₁, and b₂ are real numbers. Pearson curves are classified into 12 types, depending on the values of the parameters a, b₀, b₁, and b₂ and on the range of x. Examples of Pearson curves are the normal distribution, Student’s distribution, and the x² distribution.

Any Pearson curve y (x) is uniquely determined by its first four moments:

Pearson curves sometimes are used, on the basis of this property, in mathematical statistics for approximate representation of an unknown density p(x). For example, let there be a large series of independent observations x₁, x₂, …, x_n of a random variable x with an unknown probability density p(x). Using the method of moments, we assume that

and for an approximate representation of p(x) we select a Pearson curve y(x) such that

Pearson curves were first used to construct empirical densities by the English mathematician K. Pearson in 1894.