9.6. Normal law in the space of three dimensions.

Normal law in the space of three dimensions. General record of the normal law for a system of an arbitrary number of random variables

When investigating issues related to firing remote projectiles, one has to deal with the law of distribution of the points of rupture of a remote projectile in space. Subject to the use of conventional remote fuses, this distribution law can be considered normal.

In this we consider only the canonical form of the normal law in space:

, (9.6.1)

Where - main standard deviations.

Moving from standard deviations to probable, we have:

. (9.6.2)

When solving problems associated with firing remote projectiles, it is sometimes necessary to calculate the probability of a remote projectile rupture within a given area. . In general, this probability is triggered by a triple integral:

. (9.6.3)

The integral (9.6.3) is usually not expressed in terms of elementary functions. However, there are a number of areas, the probability of falling into which is calculated relatively simply.

1. The probability of hitting a rectangular parallelepiped with sides parallel to the main dispersion axes

Let the area is a rectangular parallelepiped bounded by abscissas ordinates and appliques (fig. 9.6.1). Probability of hitting the area obviously equal to:

. (9.6.4)

Fig. 9.6.1

2. The probability of hitting an ellipsoid of equal density

Consider an ellipsoid of equal density whose equation

.

The semi-axes of this ellipsoid are proportional to the main standard quadratic deviations:

.

Using the formula (9.6.1) for express the probability of hitting an ellipsoid :

.

Let's move from Cartesian coordinates to polar (spherical) change of variables.

(9.6.5)

The conversion Jacobian (9.6.5) is equal to:

.

Turning to the new variables, we have:

.

Integrating in parts, we get:

. (9.6.6)

3. The probability of hitting a cylindrical region with a generator parallel to one of the main scattering axes

Consider a cylindrical shine whose generator is parallel to one of the main dispersion axes (for example, ), and the guide is the contour of an arbitrary area on surface (fig. 9.6.2). Let the area limited to two planes and . Calculate the probability of hitting the area ; this is the probability of producing two events, the first of which consists in hitting a point to the area , and the second - in hit values on the plot . As magnitudes , subject to the normal law in canonical form, independent, then these two events are also independent. therefore

(9.6.7)

Fig. 9.6.2.

Probability in formula (9.6.7) can be calculated by any of the methods for calculating the probability of hitting a flat region.

The following method for calculating the probability of hitting the spatial domain is based on the formula (9.6.7) freeform: region approximately divided into a number of cylindrical areas (fig. 9.6.3), and the probability of hitting each of them is calculated by the formula (9.6.7). To use this method, it is enough to draw a series of figures representing sections of the area planes parallel to one of the coordinate planes. The probability of hitting each of them is calculated by the scattering grid.

Fig. 9.6.3.

In conclusion of this chapter, we write the general for a normal law in the space of any number of dimensions . The distribution density of such a law is:

, (9.6.8)

Where - determinant of the matrix - inverse matrix of the correlation matrix i.e. if correlation matrix

,

that

,

Where - the determinant of the correlation matrix, and - the minor of this determinant, obtained from it by striking out line and th column. notice, that

.

From the general expression (9.6.8) follow all forms of the normal law for any number of measurements and for any kinds of dependence between random variables. In particular, when (dispersion in a plane) correlation matrix is

.

Where - correlation coefficient. From here

.

Substituting the determinant of the matrix and its members in (9.6.8), we obtain the formula (9.1.1) for the normal law on the plane with which we started 9.1.