Pearson Distribution - Definition

Definition

A Pearson density p is defined to be any valid solution to the differential equation (cf. Pearson 1895, p. 381)

\frac{p'(x)}{p(x)} + \frac{a+x-\lambda}{b_2 (x-\lambda)^2 + b_1 (x-\lambda) + b_0} = 0. \qquad (1) \!

with :

According to Ord, Pearson devised the underlying form of Equation (1) on the basis of, firstly, the formula for the derivative of the logarithm of the density function of the normal distribution (which gives a linear function) and, secondly, from a recurrence relation for values in the probability mass function of the hypergeometric distribution (which yields the linear-divided-by-quadratic structure).

In Equation (1), the parameter a determines a stationary point, and hence under some conditions a mode of the distribution, since

follows directly from the differential equation.

Since we are confronted with a first order linear differential equation with variable coefficients, its solution is straightforward:

p(x) \propto \exp\left( -\!\int\!\!\frac{x-a}{b_2 x^2 + b_1 x + b_0} \,\mathrm{d}x \right). \!

The integral in this solution simplifies considerably when certain special cases of the integrand are considered. Pearson (1895, p. 367) distinguished two main cases, determined by the sign of the discriminant (and hence the number of real roots) of the quadratic function

f(x) = b_2\,x^2 + b_1\,x + b_0. \qquad (2)\!

Read more about this topic:  Pearson Distribution

Famous quotes containing the word definition:

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)

    “Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?”
    —Richard Eberhart (b. 1904)