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8.7. System of an arbitrary number of random variables

Lecture



In practice, it is often necessary to consider systems of more than two random variables. These systems are interpreted as random points or random vectors in the space of a certain number of dimensions.

We give examples.

1. The point of rupture of a remote projectile in space is characterized by three Cartesian coordinates   8.7.  System of an arbitrary number of random variables or three spherical coordinates   8.7.  System of an arbitrary number of random variables .

2. The whole   8.7.  System of an arbitrary number of random variables consecutive measurements of varying magnitude   8.7.  System of an arbitrary number of random variables - system   8.7.  System of an arbitrary number of random variables random variables   8.7.  System of an arbitrary number of random variables .

3. Shoots a burst from   8.7.  System of an arbitrary number of random variables shells. Set of coordinates   8.7.  System of an arbitrary number of random variables points of contact on the plane - system   8.7.  System of an arbitrary number of random variables random variables (abscissa and ordinate points of hitting):

  8.7.  System of an arbitrary number of random variables .

4. The initial velocity of the fragment is a random vector, characterized by three random variables: the magnitude of the velocity   8.7.  System of an arbitrary number of random variables and two corners   8.7.  System of an arbitrary number of random variables and   8.7.  System of an arbitrary number of random variables defining the direction of flight of the fragment in a spherical coordinate system.

A complete characteristic of a system of an arbitrary number of random variables is the distribution law of the system, which can be given by a distribution function or a distribution density.

System distribution function   8.7.  System of an arbitrary number of random variables random variables   8.7.  System of an arbitrary number of random variables called the probability of sharing   8.7.  System of an arbitrary number of random variables inequalities of the form   8.7.  System of an arbitrary number of random variables :

  8.7.  System of an arbitrary number of random variables . (8.7.1)

Distribution density system   8.7.  System of an arbitrary number of random variables continuous random variables called   8.7.  System of an arbitrary number of random variables -th mixed partial derivative of a function   8.7.  System of an arbitrary number of random variables taken once for each argument:

  8.7.  System of an arbitrary number of random variables . (8.7.2)

Knowing the law of distribution of the system, it is possible to determine the laws of distribution of individual quantities included in the system. The distribution function of each of the quantities included in the system is obtained if the distribution function sets all other arguments equal   8.7.  System of an arbitrary number of random variables :

  8.7.  System of an arbitrary number of random variables . (8.7.3)

If you select from the system of values   8.7.  System of an arbitrary number of random variables private system   8.7.  System of an arbitrary number of random variables , the distribution function of this system is determined by the formula

  8.7.  System of an arbitrary number of random variables . (8.7.4)

The density of distribution of each of the quantities in the system is obtained if the density of the distribution of the system is integrated in infinite limits over all other arguments:

  8.7.  System of an arbitrary number of random variables . (8.7.5)

The distribution density of the private system   8.7.  System of an arbitrary number of random variables allocated from the system   8.7.  System of an arbitrary number of random variables , is equal to:

  8.7.  System of an arbitrary number of random variables (8.7.0)

Conditional law of the distribution of the private system   8.7.  System of an arbitrary number of random variables its distribution law, calculated under the condition that the other quantities   8.7.  System of an arbitrary number of random variables took meaning   8.7.  System of an arbitrary number of random variables .

The conditional distribution density can be calculated by the formula

  8.7.  System of an arbitrary number of random variables (8.7.7)

Random variables   8.7.  System of an arbitrary number of random variables are called independent if the distribution law of each private system separated from the system   8.7.  System of an arbitrary number of random variables , does not depend on what values ​​the remaining random variables took.

The distribution density of a system of independent random variables is equal to the product of the distribution densities of the individual variables in the system:

  8.7.  System of an arbitrary number of random variables (8.7.8)

Probability of hitting a random point   8.7.  System of an arbitrary number of random variables within   8.7.  System of an arbitrary number of random variables -dimensional area   8.7.  System of an arbitrary number of random variables is expressed   8.7.  System of an arbitrary number of random variables -fold integral:

  8.7.  System of an arbitrary number of random variables . (8.7.9)

Formula (8.7.9) is essentially the basic formula for calculating the probabilities of events that are not reducible to the case diagram. Indeed, if the event we are interested in   8.7.  System of an arbitrary number of random variables does not boil down to the scheme of cases, its probability cannot be calculated directly. If it is not possible to make a sufficient number of homogeneous experiments and approximately determine the probability of an event   8.7.  System of an arbitrary number of random variables in terms of its frequency, the typical scheme for calculating the probability of an event is as follows. Transition from the scheme of events to the scheme of random variables (most often - continuous) and reduce the event   8.7.  System of an arbitrary number of random variables to the event that the system of random variables   8.7.  System of an arbitrary number of random variables will be within a certain area   8.7.  System of an arbitrary number of random variables . Then the probability of an event   8.7.  System of an arbitrary number of random variables can be calculated by the formula (8.7.9).

Example 1. An airplane is hit by a remote projectile, provided that the projectile rupture occurred not further than at a distance.   8.7.  System of an arbitrary number of random variables from the aircraft (more precisely, from the conditional point on the axis of the aircraft, taken as its center). The law of distribution of the points of rupture of a remote projectile in the coordinate system associated with the target has a density   8.7.  System of an arbitrary number of random variables . Determine the probability of hitting the aircraft.

Decision. Denoting the defeat of the aircraft by the letter   8.7.  System of an arbitrary number of random variables , we have:

  8.7.  System of an arbitrary number of random variables ,

where integration extends over the ball   8.7.  System of an arbitrary number of random variables radius   8.7.  System of an arbitrary number of random variables centered at the origin.

Example 2. A meteorite encountered on the path of an artificial Earth satellite breaks through its envelope if: 1) the angle   8.7.  System of an arbitrary number of random variables under which a meteorite meets the surface of a satellite is enclosed within certain limits.   8.7.  System of an arbitrary number of random variables ; 2) the meteorite has a weight of at least   8.7.  System of an arbitrary number of random variables (d) and 3) the relative speed of a meteorite meeting a satellite is less   8.7.  System of an arbitrary number of random variables (m / s). Meeting speed   8.7.  System of an arbitrary number of random variables meteorite weight   8.7.  System of an arbitrary number of random variables and the angle of the meeting   8.7.  System of an arbitrary number of random variables represent a system of random variables with a distribution density   8.7.  System of an arbitrary number of random variables . Find probability   8.7.  System of an arbitrary number of random variables the fact that a separate meteorite hit the satellite, breaks through its shell.

Decision. Integrating distribution density   8.7.  System of an arbitrary number of random variables on the three-dimensional region corresponding to the punching of the shell, we get:

  8.7.  System of an arbitrary number of random variables ,

Where   8.7.  System of an arbitrary number of random variables - maximum meteorite weight,   8.7.  System of an arbitrary number of random variables - Maximum meeting speed.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis