For alignment (approximation) of statistical distributions the set of various methods is used: polynomial approximation, Charlie’s series, Kramer's perturbation polynomials, Pearson’s method etc. The basic lack of polynomial approximation is formality of received distributions – the type of approximation is not connected with the nature of the random phenomenon. Charlie’s and Kramer’s methods are suitable to approximation of the distributions approached to normal. Unlike them Pearson’s method is universal enough and covers practically all known kinds of statistical distributions.
Pearson [1, 2] has suggested to use for the description of statistical distribution of random variable
(1)
where as origin of counting
Coefficients in the equation (1) can be calculated by means of the central moments . At introduction of designations they are from parities: (2) Let's enter auxiliary value: (3) Then the system of equations (2) can be written in the form of:
(2
Let's calculate a discriminant of denominator in the equation (1): . Let's designate (4) Then
The general integral of the equation (1) essentially depends on a kind of roots of a quadratic equation
also it is defined by criterion
1. At real roots of various signs. 2. At complex roots. 3. At real roots of the same sign. To each of these cases there corresponds one of the basic types of Pearson’s curves – I, IV and VI. Other nine types and normal distribution curve – their private or boundary cases. Most often in an expert there are first seven types of Pearson’s curves. On Fig. 1 the graph for definition of type of a curve on parameters [1, 2] is resulted.
Fig. 1. The graph for definition of Pearson’s curve type depending on
Let's consider the equations of Pearson’s curves of
The
κ
< 0; its equation looks like:
(5)
Exponents
(6) and at we undertake , and at – on the contrary. Coefficients are defined from formulas:
(7)
where Initial ordinate (8) Here Г (z) – gamma-function : (9)
Domain of curve
Fig. 2. Pearson’s curves of
Depending on values
three versions of curve of
1. At
its ordinates are limited (Fig. 2,
2. In case of different signs
values of density function on one of the interval
ends aspire to infinity (J-shaped curves, Fig. 2,
3. At
distribution becomes antimodal (U-shaped) – Fig. 2
The
(10) where (11)
The sign of
(12) а nd – tabulated function.
Origin of coordinates undertakes in a point
(here
– mathematical expectation of random variable
Fig. 3. Pearson’s curve of
The
κ
> 1 and is described by the equation:
(13) Here (14) (15) And the condition should be satisfied. Origin of coordinates undertakes in a point and initial ordinate (16)
The curve lies on an interval from
Fig. 4. Pearson’s curves of
The following group of Pearson’s curves corresponds to private values of criterion
The
(17)
and
(18) Mode exists at
Special case of Pearson’s curve of
(19) Origin of coordinates – in a point .
κ
= 1. Its equation looks like:
(20) Here (21)
Function
Fig. 5. Pearson’s curve of
At
The
(22) where (23)
Coefficient
Origin of coordinates corresponds to average value of statistical distribution. The curve of
The
(24) where (25) Coefficient Origin of coordinates corresponds to average value of a random variable.
The curve of
The
(26) turns out at and Origin of coordinates corresponds to observed average value. The basic properties of normal distribution are well-known. |

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>> Applied Mathematics
>> Mathematical Statistics
>> Treatment of Experiment Results
>> Alignment of statistical distributions. Pearson’s curves