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9.6. Normal law in the space of three dimensions.

Lecture



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   9.6.  Normal law in the space of three dimensions. we consider only the canonical form of the normal law in space:

  9.6.  Normal law in the space of three dimensions. , (9.6.1)

Where   9.6.  Normal law in the space of three dimensions. - main standard deviations.

Moving from standard deviations to probable, we have:

  9.6.  Normal law in the space of three dimensions. . (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.   9.6.  Normal law in the space of three dimensions. . In general, this probability is triggered by a triple integral:

  9.6.  Normal law in the space of three dimensions. . (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   9.6.  Normal law in the space of three dimensions. is a rectangular parallelepiped bounded by abscissas   9.6.  Normal law in the space of three dimensions. ordinates   9.6.  Normal law in the space of three dimensions. and appliques   9.6.  Normal law in the space of three dimensions. (fig. 9.6.1). Probability of hitting the area   9.6.  Normal law in the space of three dimensions. obviously equal to:

  9.6.  Normal law in the space of three dimensions. . (9.6.4)

  9.6.  Normal law in the space of three dimensions.

Fig. 9.6.1

2. The probability of hitting an ellipsoid of equal density

Consider an ellipsoid of equal density   9.6.  Normal law in the space of three dimensions. whose equation

  9.6.  Normal law in the space of three dimensions. .

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

  9.6.  Normal law in the space of three dimensions. .

Using the formula (9.6.1) for   9.6.  Normal law in the space of three dimensions. express the probability of hitting an ellipsoid   9.6.  Normal law in the space of three dimensions. :

  9.6.  Normal law in the space of three dimensions. .

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

  9.6.  Normal law in the space of three dimensions. (9.6.5)

The conversion Jacobian (9.6.5) is equal to:

  9.6.  Normal law in the space of three dimensions. .

Turning to the new variables, we have:

  9.6.  Normal law in the space of three dimensions. .

Integrating in parts, we get:

  9.6.  Normal law in the space of three dimensions. . (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   9.6.  Normal law in the space of three dimensions. whose generator is parallel to one of the main dispersion axes (for example,   9.6.  Normal law in the space of three dimensions. ), and the guide is the contour of an arbitrary area   9.6.  Normal law in the space of three dimensions. on surface   9.6.  Normal law in the space of three dimensions. (fig. 9.6.2). Let the area   9.6.  Normal law in the space of three dimensions. limited to two planes   9.6.  Normal law in the space of three dimensions. and   9.6.  Normal law in the space of three dimensions. . Calculate the probability of hitting the area   9.6.  Normal law in the space of three dimensions. ; this is the probability of producing two events, the first of which consists in hitting a point   9.6.  Normal law in the space of three dimensions. to the area   9.6.  Normal law in the space of three dimensions. , and the second - in hit values   9.6.  Normal law in the space of three dimensions. on the plot   9.6.  Normal law in the space of three dimensions. . As magnitudes   9.6.  Normal law in the space of three dimensions. , subject to the normal law in canonical form, independent, then these two events are also independent. therefore

  9.6.  Normal law in the space of three dimensions. (9.6.7)

  9.6.  Normal law in the space of three dimensions.

Fig. 9.6.2.

Probability   9.6.  Normal law in the space of three dimensions. 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)   9.6.  Normal law in the space of three dimensions. freeform: region   9.6.  Normal law in the space of three dimensions. approximately divided into a number of cylindrical areas   9.6.  Normal law in the space of three dimensions. (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   9.6.  Normal law in the space of three dimensions. planes parallel to one of the coordinate planes. The probability of hitting each of them is calculated by the scattering grid.

  9.6.  Normal law in the space of three dimensions.

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   9.6.  Normal law in the space of three dimensions. . The distribution density of such a law is:

  9.6.  Normal law in the space of three dimensions. , (9.6.8)

Where   9.6.  Normal law in the space of three dimensions. - determinant of the matrix   9.6.  Normal law in the space of three dimensions. - inverse matrix of the correlation matrix   9.6.  Normal law in the space of three dimensions. i.e. if correlation matrix

  9.6.  Normal law in the space of three dimensions. ,

that

  9.6.  Normal law in the space of three dimensions. ,

Where   9.6.  Normal law in the space of three dimensions. - the determinant of the correlation matrix, and   9.6.  Normal law in the space of three dimensions. - the minor of this determinant, obtained from it by striking out   9.6.  Normal law in the space of three dimensions. line and   9.6.  Normal law in the space of three dimensions. th column. notice, that

  9.6.  Normal law in the space of three dimensions. .

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   9.6.  Normal law in the space of three dimensions. (dispersion in a plane) correlation matrix is

  9.6.  Normal law in the space of three dimensions. .

Where   9.6.  Normal law in the space of three dimensions. - correlation coefficient. From here

  9.6.  Normal law in the space of three dimensions.

  9.6.  Normal law in the space of three dimensions. .

Substituting the determinant of the matrix   9.6.  Normal law in the space of three dimensions. 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   9.6.  Normal law in the space of three dimensions. 9.1.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis