Normal distribution

The main types of distribution used in forestry

Table of contents

The main types of distribution used in forestry . one

1. Algorithm for determining the law of distribution of an empirical series . one

2. Normal distribution . 2

3. Logarithmically normal (lognormal) distribution . 5

4. Weibull distribution . 6

Algorithm for determining the law of distribution of an empirical series

A random variable can be specified as a distribution series (numerically or graphically). Figure 1 shows the distribution of a random variable in the form of a frequency polygon. According to this graph, we observe certain patterns in the distribution of the option: the closer the value of the option to , the greater their frequency, and vice versa, the farther, the less often they occur.

Rice. 1. Polygon of frequencies

In practice, distribution theory is applied in the following areas:

Establishment of the law of distribution of the studied quantity in the sample;

Obtaining an analytical type of distribution.

On Fig.2. an algorithm for establishing the law of distribution of a random variable is presented.

Fig.2. Scheme for determining the law of distribution

There are more than two dozen different distribution laws; in forestry practice, the following are mainly used: normal, logarithmically normal, Poisson, etc.

Normal distribution

A type of continuous distribution, discovered by A. Moivre in 1733 (England), then rediscovered by K. Gauss (1809), P. Laplace (1812), therefore it is called the Laplace-Gauss distribution.

The distribution of a random variable is considered normal if the skewness and kurtosis coefficients are equal to zero. This is a strict condition that is rarely observed in nature:

(one)

Therefore, a less stringent condition is used when the skewness and kurtosis coefficients do not exceed their double errors:

(2)

The density of the normal distribution ƒ(x) has the form:

ƒ(x) = exp ; (3)

Distribution function Ф(x) (cumulative probability function)

Ф(x) = dx , (4)

where – mean,

– standard deviation,

e is Euler’s constant, the base of natural logarithms, equal to 2.718.

The graph (Fig. 3) shows a graph of the density of the normal distribution ƒ (x) – this is a symmetrical bell-shaped curve. Its shape depends on the size , which is a scale parameter, the position depends on the second parameter − . Those. the normal distribution is a two-parameter distribution whose parameters are: and . The curve has:

one maximum ,

two inflection points at a distance of ±σ from .

Fig.3. Normal distribution curve with different parameters

The calculation of the distribution density according to formula (1) is quite laborious. Therefore, tables of values ƒ(x) and F(x) are compiled for the standard distribution, where it is assumed that =0 and σ =1 . Variables have also been changed:

t= (5)

then the density and normalized normal distribution function are:

ƒ(t)= exp(- ) (6);

Ф(t)= (- )dt (7).

In App. 3 shows tables of the standard normal distribution.

Properties of the normal distribution

1. All options lie in the interval , in other words, with a probability of 1 (100%), a new option is expected to appear within .

2. To the left and to the right of the arithmetic mean lies a 50% variant, i.e., with a probability of 0.5 (50%), it is possible to predict the appearance of a new variant to the left or to the right of the mean.

3. In the interval from ±1 68.3% of all options lie, hence with a probability of 0.683 (68.3%) it is possible to predict the emergence of a new option at a distance of ±1 from the middle – the rule of one sigma ) (Fig. 4, a);

4. Between ±1.96 lie 95% option. This allows us to assume with a 95% probability that the new option will be in the interval ±1.96 (rounded ±2the rule of two sigma ) (Fig. 4, b);

5. With a probability of 0.99 (99%), the value of the new option will be within ±2.58 ( rule of three sigma ).) and with a probability of 0.999 – in the interval ±3.3 (Fig.4, c).

Rice. 4. Properties of the normal distribution curve

Comparison of theoretical frequencies with empirical ones in order to establish whether this distribution differs from the normal one is carried out by several methods, for example, using the Kolmogorov-Smirnov criterion, χ 2 distribution.

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