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4.2. General repetition theorem

Lecture



The particular theorem on repetition of experiments concerns the case when the probability of an event   4.2.  General repetition theorem in all experiments is the same. In practice, one often encounters a more complex case, when experiments are carried out in different conditions, and the probability of an event varies from experience to experience. For example, if a series of shots are made in varying conditions (for example, with varying range), then the probability of hitting from shot to shot may change noticeably.

The method of calculating the probability of a given number of occurrences of an event in such conditions gives a general theorem on the repetition of experiments.

Let n independent experiments be performed, in each of which some event may or may not appear.   4.2.  General repetition theorem , with the probability of occurrence   4.2.  General repetition theorem in i-th experience is equal   4.2.  General repetition theorem and the probability of non-occurrence   4.2.  General repetition theorem . Required to find the probability   4.2.  General repetition theorem that as a result   4.2.  General repetition theorem experiences event   4.2.  General repetition theorem will appear exactly   4.2.  General repetition theorem time.

Denote as before   4.2.  General repetition theorem event consisting in that event   4.2.  General repetition theorem will appear   4.2.  General repetition theorem once in   4.2.  General repetition theorem experiences. Still imagine   4.2.  General repetition theorem as the sum of the products of elementary events:

  4.2.  General repetition theorem

and in each of the works of the event   4.2.  General repetition theorem enters   4.2.  General repetition theorem time event   4.2.  General repetition theorem -   4.2.  General repetition theorem time. The number of such combinations will still be   4.2.  General repetition theorem , but the combinations themselves will be already unequal.

Applying the addition theorem and the multiplication theorem for independent events, we obtain:

  4.2.  General repetition theorem

those. the required probability is equal to the sum of all possible products in which the letters   4.2.  General repetition theorem with different indexes are included   4.2.  General repetition theorem times and letters   4.2.  General repetition theorem with different indexes   4.2.  General repetition theorem time.

In order to mechanically compose all possible works from   4.2.  General repetition theorem letters   4.2.  General repetition theorem and   4.2.  General repetition theorem letters   4.2.  General repetition theorem with different indices, apply the following formal method. We make the product   4.2.  General repetition theorem binomials:

  4.2.  General repetition theorem

or shorter

  4.2.  General repetition theorem ,

Where   4.2.  General repetition theorem - arbitrary parameter.

Let us set ourselves the goal of finding in this product of binomials a coefficient with   4.2.  General repetition theorem . To do this, multiply the binomials and produce a cast of such terms. Obviously, every member containing   4.2.  General repetition theorem , will have as a coefficient the product   4.2.  General repetition theorem letters   4.2.  General repetition theorem with some indices and   4.2.  General repetition theorem letters   4.2.  General repetition theorem and after bringing such members into account   4.2.  General repetition theorem will be the sum of all possible works of this type. Consequently, the method of compiling this coefficient completely coincides with the method of calculating the probability   4.2.  General repetition theorem in the problem of repeating experiments.

Function   4.2.  General repetition theorem whose decomposition is in powers of the parameter   4.2.  General repetition theorem gives probability coefficients   4.2.  General repetition theorem , is called the generating probability function   4.2.  General repetition theorem , or simply generating function.

Using the concept of a generating function, we can formulate a theorem on the repetition of experiments in the following form.

Probability that an event   4.2.  General repetition theorem at   4.2.  General repetition theorem independent experiences will appear exactly   4.2.  General repetition theorem times equal to the coefficient at   4.2.  General repetition theorem in the expression of the generating function:

  4.2.  General repetition theorem ,

Where   4.2.  General repetition theorem - probability of occurrence   4.2.  General repetition theorem in the i-th experience   4.2.  General repetition theorem .

The above formulation of the general theorem on the repetition of experiments, in contrast to the particular theorem, does not give an explicit expression for the probability   4.2.  General repetition theorem . Such an expression, in principle, can be written, but it is too complex, and we will not give it. However, without resorting to such an explicit expression, it is still possible to write the general theorem on the repetition of experiments in the form of one formula:

  4.2.  General repetition theorem . (4.2.1)

The left and right sides of equality (4.2.1) are the same generating function.   4.2.  General repetition theorem , only to the left it is written as a monomial, and to the right - as a polynomial. Opening the brackets on the left side and performing the cast of such terms, we get all the probabilities:

  4.2.  General repetition theorem

as coefficients, respectively, at zero, first, etc. degrees   4.2.  General repetition theorem .

Obviously, the particular theorem on the repetition of experiments follows from the general at

  4.2.  General repetition theorem

In this case, the generating function refers to   4.2.  General repetition theorem degree of binomial   4.2.  General repetition theorem :

  4.2.  General repetition theorem .

Opening this expression according to the formula of the binomial, we have:

  4.2.  General repetition theorem ,

whence formula (4.1.1) follows.

Note that, both in general and in the particular case, the sum of all probabilities   4.2.  General repetition theorem equals one:

  4.2.  General repetition theorem . (4.2.2)

This follows, first of all, from the fact that events   4.2.  General repetition theorem form a complete group of incompatible events. Formally, equality (4.2.2) can be achieved by assuming in the general formula (4.2.1)   4.2.  General repetition theorem .

In many cases, the practice, besides the probability   4.2.  General repetition theorem smooth   4.2.  General repetition theorem occurrences of event A, we must consider the probability of not less than   4.2.  General repetition theorem occurrences of the event A.

Denote   4.2.  General repetition theorem an event consisting in that event A appears at least   4.2.  General repetition theorem times, and the probability of an event   4.2.  General repetition theorem denote   4.2.  General repetition theorem . Obviously

  4.2.  General repetition theorem ,

from which, by the addition theorem,

  4.2.  General repetition theorem ,

or shorter

  4.2.  General repetition theorem . (4.2.3)

When calculating   4.2.  General repetition theorem it is often more convenient not to use directly the formula (4.2.3), but to go to the opposite event and calculate the probability   4.2.  General repetition theorem according to the formula

  4.2.  General repetition theorem . (4.2.4)

Example 1. 4 independent shots are fired at the same target from different distances; the probabilities of hitting with these shots are respectively

  4.2.  General repetition theorem .

Find the probabilities of no, one, two, three, or four hits:

  4.2.  General repetition theorem .

Decision. We make the generating function:

  4.2.  General repetition theorem

from where

  4.2.  General repetition theorem .

Example 2. 4 independent shots are made under the same conditions, and the probability of hitting p is the average of the probabilities   4.2.  General repetition theorem previous example:

  4.2.  General repetition theorem .

Find probabilities

  4.2.  General repetition theorem .

Decision. By the formula (4.1.1) we have:

  4.2.  General repetition theorem

Example 3. There are 5 stations with which communication is supported. From time to time the connection is interrupted due to atmospheric noise. Due to the distance of the stations from each other, the interruption of communication with each of them occurs independently of the others with a probability   4.2.  General repetition theorem . Find the probability that at a given time there is a connection with no more than two stations.

Decision. The event in question, is reduced to the fact that communication will be broken no less than the stirrup stations. By the formula (4.2.3) we get:

  4.2.  General repetition theorem

Example 4. A system of radar stations monitors a group of objects consisting of 10 units. Each of the objects can be (independently of the others) lost with a probability of 0.1. find the probability that at least one of the objects will be lost.

Decision. The probability of losing at least one object   4.2.  General repetition theorem could be found by the formula

  4.2.  General repetition theorem ,

but it is much easier to use the probability of the opposite event - not a single object is lost - and subtract it from one:

  4.2.  General repetition theorem .

Example 5. The device consists of 8 homogeneous elements, but can work in the presence in good condition of at least 6 of them. Each of the elements during the operation of the device   4.2.  General repetition theorem fails independently of others with a probability of 0.2. Find the probability that the device fails in time.   4.2.  General repetition theorem .

Decision. For failure of the device requires the failure of at least two of the eight elements. By the formula (4.2.4) we have:

  4.2.  General repetition theorem .

Example 6. There are 4 independent shots from the aircraft on the aircraft. The probability of hitting each shot is 0.3. For defeat (failure) of the plane, two hits are known to be sufficient; with one plane is hit with a probability of 0.6. Find the probability that the plane will be hit.

Decision. The problem is solved by the formula of total probability. Hypotheses could be considered.

  4.2.  General repetition theorem - 1 projectile hit the plane,

  4.2.  General repetition theorem - 2 missiles hit the plane,

  4.2.  General repetition theorem - 3 missiles hit the plane,

  4.2.  General repetition theorem - 4 missiles hit the plane

and to find the probability of an A event - the destruction of an aircraft - using these four hypotheses. However, it is much easier to consider only two hypotheses:

  4.2.  General repetition theorem - not a single shell hit the plane,

  4.2.  General repetition theorem - 1 projectile hit the plane,

and calculate the probability of an event   4.2.  General repetition theorem - non-impact aircraft:

  4.2.  General repetition theorem

We have:

  4.2.  General repetition theorem

Consequently,

  4.2.  General repetition theorem ,

from where

  4.2.  General repetition theorem .


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis