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9.3. The probability of hitting a rectangle with sides parallel to the main axes of dispersion

Lecture



Let random point   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion on the plane is subject to normal law

  9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion (9.3.1)

  9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion

Fig. 9.3.1.

In this case, the main axes of dispersion are parallel to the coordinate axes and the values   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion and   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion are independent.

It is required to calculate the probability of hitting a random point.   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion into a rectangle   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion whose sides are parallel to the coordinate axes   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion , and, therefore, the main axes of dispersion (Fig. 9.3.1). According to the general formula (8.3.4) we have:

  9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion .

whence, applying the formula (6.3.3) for the probability of hitting the site, we find:

  9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion , (9.3.2)

Where   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion - normal distribution function.

If the normal law on a plane is given in canonical form, then   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion , and the formula (9.3.2) takes the form

  9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion . (9.3.3)

If the sides of the rectangle are not parallel to the coordinate axes, then formulas (9.3.2) and (9.3.3) are no longer applicable. Only with circular scattering, the probability of hitting a rectangle of any orientation is calculated by the formula (9.3.2) or (9.3.3).

Formulas (9.3.2) and (9.3.3) are widely used in calculating the probabilities of hitting the target: rectangular, close to rectangular, composed of rectangles, or approximately replaced by those.

Example. Shooting from a plane at a rectangular shield size   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion lying on the ground horizontally. Main probable deviations: in the longitudinal direction   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion in the lateral direction   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion . Aiming - in the center of the target, approach - along the target. Due to the discrepancy between the shooting range and the actual shooting range, the average point of impact shifts towards undershoot by   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion . Find the probability of hitting the target with one shot.

  9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion

Fig. 9.3.2.

Decision. In the drawing (fig. 9.3.2) we put the target, the aiming point (etc.) and the center of dispersion (c. P.). Through ts.r. draw the main axis of dispersion: in the direction of flight and perpendicular to it.

Let's go from the main probable deviations   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion and   9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion to the main mean squares:

  9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion .

By the formula (9.3.3) we have:

  9.3.  The probability of hitting a rectangle with sides parallel to the main axes of dispersion


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis