You get a bonus - 1 coin for daily activity. Now you have 1 coin

13.3. The law of large numbers (Chebyshev theorem)

Lecture



In this   13.3.  The law of large numbers (Chebyshev theorem) we prove one of the simplest, but at the same time the most important forms of the law of large numbers - the Chebyshev theorem. This theorem establishes a connection between the arithmetic mean of the observed values ​​of a random variable and its expectation.

First we solve the following auxiliary problem.

There is a random variable   13.3.  The law of large numbers (Chebyshev theorem) with mathematical expectation   13.3.  The law of large numbers (Chebyshev theorem) and variance   13.3.  The law of large numbers (Chebyshev theorem) . Above this value is   13.3.  The law of large numbers (Chebyshev theorem) independent experiments and calculates the arithmetic average of all observed values ​​of   13.3.  The law of large numbers (Chebyshev theorem) . It is required to find the numerical characteristics of this arithmetic average - the expectation and variance - and find out how they change with increasing   13.3.  The law of large numbers (Chebyshev theorem) .

Denote:

  13.3.  The law of large numbers (Chebyshev theorem) - value   13.3.  The law of large numbers (Chebyshev theorem) in the first experience;

  13.3.  The law of large numbers (Chebyshev theorem) - value   13.3.  The law of large numbers (Chebyshev theorem) in the second experiment, etc.

Obviously, a set of values   13.3.  The law of large numbers (Chebyshev theorem) represents   13.3.  The law of large numbers (Chebyshev theorem) independent random variables, each of which is distributed according to the same law as the quantity itself   13.3.  The law of large numbers (Chebyshev theorem) . Consider the arithmetic average of these values:

  13.3.  The law of large numbers (Chebyshev theorem) .

Random value   13.3.  The law of large numbers (Chebyshev theorem) there is a linear function of independent random variables   13.3.  The law of large numbers (Chebyshev theorem) . Find the expectation and variance of this value. According to the rules   13.3.  The law of large numbers (Chebyshev theorem) 10 to determine the numerical characteristics of linear functions, we obtain:

  13.3.  The law of large numbers (Chebyshev theorem) ;

  13.3.  The law of large numbers (Chebyshev theorem) .

So, the expected value of   13.3.  The law of large numbers (Chebyshev theorem) does not depend on the number of experiences   13.3.  The law of large numbers (Chebyshev theorem) and equal to the expected value of the observed value   13.3.  The law of large numbers (Chebyshev theorem) ; as for the variance of magnitude   13.3.  The law of large numbers (Chebyshev theorem) then it decreases unlimitedly with an increase in the number of experiments and with a sufficiently large   13.3.  The law of large numbers (Chebyshev theorem) can be made arbitrarily small. We see that the arithmetic mean is a random variable with an arbitrarily small variance and, with a large number of experiments, behaves almost as not random.

Chebyshev's theorem establishes in exact quantitative form this property of stability of the arithmetic mean. It is formulated as follows:

With a sufficiently large number of independent experiments, the arithmetic mean of the observed values ​​of a random variable converges in probability to its expected value.

We write the Chebyshev theorem as a formula. For this we recall the meaning of the term "converges in probability." It is said that a random variable   13.3.  The law of large numbers (Chebyshev theorem) converges in probability to value   13.3.  The law of large numbers (Chebyshev theorem) if increasing   13.3.  The law of large numbers (Chebyshev theorem) probability that   13.3.  The law of large numbers (Chebyshev theorem) and   13.3.  The law of large numbers (Chebyshev theorem) will be arbitrarily close, unlimitedly approaching unity, which means that with a sufficiently large   13.3.  The law of large numbers (Chebyshev theorem)

  13.3.  The law of large numbers (Chebyshev theorem) ,

Where   13.3.  The law of large numbers (Chebyshev theorem) - arbitrarily small positive numbers.

We write in a similar form the Chebyshev theorem. She claims that while increasing   13.3.  The law of large numbers (Chebyshev theorem) average   13.3.  The law of large numbers (Chebyshev theorem) converges in probability to   13.3.  The law of large numbers (Chebyshev theorem) i.e.

  13.3.  The law of large numbers (Chebyshev theorem) . (13.3.1)

Let us prove this inequality.

Evidence. Above it was shown that

  13.3.  The law of large numbers (Chebyshev theorem)

has numeric characteristics

  13.3.  The law of large numbers (Chebyshev theorem) ;   13.3.  The law of large numbers (Chebyshev theorem) .

Apply to random value   13.3.  The law of large numbers (Chebyshev theorem) Chebyshev's inequality, believing   13.3.  The law of large numbers (Chebyshev theorem) :

  13.3.  The law of large numbers (Chebyshev theorem) .

No matter how small the number   13.3.  The law of large numbers (Chebyshev theorem) you can take   13.3.  The law of large numbers (Chebyshev theorem) so big that inequality holds

  13.3.  The law of large numbers (Chebyshev theorem)

Where   13.3.  The law of large numbers (Chebyshev theorem) - arbitrarily small number.

Then

  13.3.  The law of large numbers (Chebyshev theorem) ,

whence, moving to the opposite event, we have:

  13.3.  The law of large numbers (Chebyshev theorem) ,

Q.E.D.


Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis