Lecture
Chebyshev's theorem can easily be generalized to a more complicated case, namely, when the distribution law of a random variable from experience to experience does not remain the same, but changes. Then instead of the arithmetic mean of the observed values of the same value with constant expectation and variance we are dealing with arithmetic average different random variables, with different mathematical expectations and variances. It turns out that even in this case, if certain conditions are met, the arithmetic average is stable and converges in probability to a certain non-random value.
The generalized Chebyshev theorem is formulated as follows. If a
-
independent random variables with mathematical expectations
and dispersions
and if all variances are bounded above by the same number :
,
then with increasing arithmetic average of observed values converges in probability to the arithmetic mean of their mathematical expectations.
We write this theorem as a formula. Let be - arbitrarily small positive numbers. Then with a sufficiently large
. (13.4.1)
Evidence. Consider the value
.
Her expectation is:
,
and the variance
.
Apply to value Chebyshev's inequality:
,
or
. (13.4.2)
Replace the right side of the inequality (13.4.2) each of the quantities larger value . Then the inequality will only intensify:
.
However small , can choose so large that inequality holds
;
then
,
whence, passing to the opposite event, we obtain the required inequality (13.4.1).
The law of large numbers can also be extended to dependent random variables. The generalization of the law of large numbers to the case of dependent random variables belongs to A. A. Markov.
Markov theorem. If there are dependent random variables and if at
,
the arithmetic mean of the observed values of random variables converges in probability to the arithmetic mean of their mathematical expectations. Evidence. Consider the value
.
Obviously
.
Apply to value Chebyshev's inequality:
.
As by the condition of the theorem, then at a sufficiently large
,
or, moving to the opposite event,
,
Q.E.D.
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Probability theory. Mathematical Statistics and Stochastic Analysis
Terms: Probability theory. Mathematical Statistics and Stochastic Analysis