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9.4. The probability of hitting an ellipse

Lecture



Among the few flat figures, the probability of hitting which can be calculated in the final form, belongs to the ellipse of dispersion (an ellipse of equal density).

Let the normal law on the plane be given in canonical form:

  9.4.  The probability of hitting an ellipse . (9.4.1)

Consider the scattering ellipse   9.4.  The probability of hitting an ellipse whose equation

  9.4.  The probability of hitting an ellipse ,

where is the parameter   9.4.  The probability of hitting an ellipse is the ratio of the semiaxes of the dispersion ellipse to the main standard quadratic deviations. According to the general formula (8.3.3) we have:

  9.4.  The probability of hitting an ellipse . (9.4.2)

We make in the integral (9.4.2) the change of variables

  9.4.  The probability of hitting an ellipse .

This substitution ellipse   9.4.  The probability of hitting an ellipse converted to a circle   9.4.  The probability of hitting an ellipse radius   9.4.  The probability of hitting an ellipse . Consequently,

  9.4.  The probability of hitting an ellipse . (9.4.3)

We proceed in the integral (9.4.3) from the Cartesian coordinate system to the polar one, setting

  9.4.  The probability of hitting an ellipse . (9.4.4)

The conversion Jacobian (9.4.4) is   9.4.  The probability of hitting an ellipse . Making the change of variables, we get:

  9.4.  The probability of hitting an ellipse .

Thus, the probability of hitting a random point in the dispersion ellipse, whose semi-axes are equal   9.4.  The probability of hitting an ellipse standard deviations equal to:

  9.4.  The probability of hitting an ellipse . (9.4.5)

As an example, we find the probability of hitting a random point distributed according to the normal law on the plane   9.4.  The probability of hitting an ellipse in the unit ellipse of dispersion, whose semi-axes are equal to the mean square deviations:

  9.4.  The probability of hitting an ellipse .

For such an ellipse   9.4.  The probability of hitting an ellipse . We have:

  9.4.  The probability of hitting an ellipse

Using table 2 of the application, we find:

  9.4.  The probability of hitting an ellipse .

Formula (9.4.5) is most often used to calculate the probability of hitting a circle with circular scattering.

Example. On the path of a fast moving small size target   9.4.  The probability of hitting an ellipse put a fragmentation field in the form of a flat disk of radius   9.4.  The probability of hitting an ellipse . Inside the disk, the density of the fragments is constant and equal to   9.4.  The probability of hitting an ellipse . If the target is covered with a disk, then the number of fragments falling into it can be considered distributed according to the Poisson law. Due to the smallness of the target, it can be considered as a point and it can be considered that it is either completely covered by a fragmentation field (if its center falls within a fragmentation circle), or not at all covered (if its center does not fall within a circle). A hit of a shard guarantees defeat of the target. When aiming center circle   9.4.  The probability of hitting an ellipse seek to combine in the plane   9.4.  The probability of hitting an ellipse with origin   9.4.  The probability of hitting an ellipse (target center), but due to point errors   9.4.  The probability of hitting an ellipse scatters about   9.4.  The probability of hitting an ellipse (fig. 9.4.1). The law of dispersion is normal, the dispersion is circular,   9.4.  The probability of hitting an ellipse . Determine the probability of hitting the target.   9.4.  The probability of hitting an ellipse .

  9.4.  The probability of hitting an ellipse

Fig. 9.4.1

Decision. For a target to be hit by shrapnel, it is necessary to combine two events: 1) hitting a target (point   9.4.  The probability of hitting an ellipse ) in a fragmentation field (circle of radius   9.4.  The probability of hitting an ellipse ) and 2) defeat the target, provided that the hit occurred.

The probability of hitting a target in a circle is obviously equal to the probability that the center of the circle (a random point   9.4.  The probability of hitting an ellipse ) gets into a circle of radius   9.4.  The probability of hitting an ellipse , described around the origin. Apply the formula (9.4.5). We have:

  9.4.  The probability of hitting an ellipse .

The probability of hitting a target in a fragmentation field is:

  9.4.  The probability of hitting an ellipse .

Next, we find the probability of hitting the target.   9.4.  The probability of hitting an ellipse provided it is covered with a fragmentation disk. Average number of shards   9.4.  The probability of hitting an ellipse that fall into a covered field target is equal to the product of the target area and the density of the fragment field:

  9.4.  The probability of hitting an ellipse .

Conditional probability of hitting the target   9.4.  The probability of hitting an ellipse there is nothing like the probability of hitting at least one fragment in it. Using the formula (5.9.5) of chapter 5, we have:

  9.4.  The probability of hitting an ellipse .

The probability of hitting a target is:

  9.4.  The probability of hitting an ellipse .

Let us use the formula (9.4.5) for the probability of hitting the circle in order to derive one important distribution for practice: the so-called Rayleigh distribution.

Consider on the plane   9.4.  The probability of hitting an ellipse (fig. 9.4.2) random point   9.4.  The probability of hitting an ellipse scattering around the origin   9.4.  The probability of hitting an ellipse according to a circular normal law with a standard deviation   9.4.  The probability of hitting an ellipse . Find the distribution law of a random variable   9.4.  The probability of hitting an ellipse - distance from point   9.4.  The probability of hitting an ellipse to the origin, i.e. lengths of a random vector with components   9.4.  The probability of hitting an ellipse .

  9.4.  The probability of hitting an ellipse

Fig. 9.4.2.

We first find the distribution function   9.4.  The probability of hitting an ellipse magnitudes   9.4.  The probability of hitting an ellipse . By definition

  9.4.  The probability of hitting an ellipse .

This is nothing but the probability of hitting a random point.   9.4.  The probability of hitting an ellipse inside a circle of radius   9.4.  The probability of hitting an ellipse (fig. 9.4.2). By the formula (9.4.5), this probability is equal to:

  9.4.  The probability of hitting an ellipse ,

Where   9.4.  The probability of hitting an ellipse i.e.

  9.4.  The probability of hitting an ellipse . (9.4.6)

This expression of the distribution function makes sense only for positive values.   9.4.  The probability of hitting an ellipse ; with negative   9.4.  The probability of hitting an ellipse need to put   9.4.  The probability of hitting an ellipse .

Differentiating the distribution function   9.4.  The probability of hitting an ellipse by   9.4.  The probability of hitting an ellipse let's find the distribution density

  9.4.  The probability of hitting an ellipse (9.4.7)

The Rayleigh law (9.4.7) is found in different areas of practice in shooting, radio engineering, electrical engineering, etc.

Function graph   9.4.  The probability of hitting an ellipse (density Rayleigh law) is shown in Figure 9.4.3.

  9.4.  The probability of hitting an ellipse

Fig. 9.4.3

Find the numerical characteristics of the value   9.4.  The probability of hitting an ellipse distributed according to the Rayleigh law, namely: its fashion   9.4.  The probability of hitting an ellipse and mathematical expectation   9.4.  The probability of hitting an ellipse . In order to find the mode — the abscissa of the point at which the probability density is maximum, we differentiate   9.4.  The probability of hitting an ellipse and equate the derivative to zero:

  9.4.  The probability of hitting an ellipse .

The root of this equation is the desired mode.

  9.4.  The probability of hitting an ellipse . (9.4.8)

Thus, the most probable distance value   9.4.  The probability of hitting an ellipse random point   9.4.  The probability of hitting an ellipse from the origin is equal to the mean square deviation of dispersion.

Expected value   9.4.  The probability of hitting an ellipse find the formula

  9.4.  The probability of hitting an ellipse .

Replacing a variable

  9.4.  The probability of hitting an ellipse .

we will receive:

  9.4.  The probability of hitting an ellipse .

Integrating in parts, we find the expectation of the distance   9.4.  The probability of hitting an ellipse :

  9.4.  The probability of hitting an ellipse . (9.4.9)


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis