Lecture
In this 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 with mathematical expectation and variance . Above this value is independent experiments and calculates the arithmetic average of all observed values of . It is required to find the numerical characteristics of this arithmetic average - the expectation and variance - and find out how they change with increasing .
Denote:
- value in the first experience;
- value in the second experiment, etc.
Obviously, a set of values represents independent random variables, each of which is distributed according to the same law as the quantity itself . Consider the arithmetic average of these values:
.
Random value there is a linear function of independent random variables . Find the expectation and variance of this value. According to the rules 10 to determine the numerical characteristics of linear functions, we obtain:
;
.
So, the expected value of does not depend on the number of experiences and equal to the expected value of the observed value ; as for the variance of magnitude then it decreases unlimitedly with an increase in the number of experiments and with a sufficiently large 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 converges in probability to value if increasing probability that and will be arbitrarily close, unlimitedly approaching unity, which means that with a sufficiently large
,
Where - arbitrarily small positive numbers.
We write in a similar form the Chebyshev theorem. She claims that while increasing average converges in probability to i.e.
. (13.3.1)
Let us prove this inequality.
Evidence. Above it was shown that
has numeric characteristics
; .
Apply to random value Chebyshev's inequality, believing :
.
No matter how small the number you can take so big that inequality holds
Where - arbitrarily small number.
Then
,
whence, moving to the opposite event, we have:
,
Q.E.D.
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Probability theory. Mathematical Statistics and Stochastic Analysis
Terms: Probability theory. Mathematical Statistics and Stochastic Analysis