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, …, xn 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.