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

13.4. The generalized Chebyshev theorem. Markov's theorem

Lecture



Chebyshev's theorem can easily be generalized to a more complicated case, namely, when the distribution law of a random variable   13.4.  The generalized Chebyshev theorem.  Markovs theorem 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   13.4.  The generalized Chebyshev theorem.  Markovs theorem with constant expectation and variance we are dealing with arithmetic average   13.4.  The generalized Chebyshev theorem.  Markovs theorem 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

  13.4.  The generalized Chebyshev theorem.  Markovs theorem -

independent random variables with mathematical expectations

  13.4.  The generalized Chebyshev theorem.  Markovs theorem

and dispersions

  13.4.  The generalized Chebyshev theorem.  Markovs theorem

and if all variances are bounded above by the same number   13.4.  The generalized Chebyshev theorem.  Markovs theorem :

  13.4.  The generalized Chebyshev theorem.  Markovs theorem   13.4.  The generalized Chebyshev theorem.  Markovs theorem ,

then with increasing   13.4.  The generalized Chebyshev theorem.  Markovs theorem arithmetic average of observed values   13.4.  The generalized Chebyshev theorem.  Markovs theorem converges in probability to the arithmetic mean of their mathematical expectations.

We write this theorem as a formula. Let be   13.4.  The generalized Chebyshev theorem.  Markovs theorem - arbitrarily small positive numbers. Then with a sufficiently large   13.4.  The generalized Chebyshev theorem.  Markovs theorem

  13.4.  The generalized Chebyshev theorem.  Markovs theorem . (13.4.1)

Evidence. Consider the value

  13.4.  The generalized Chebyshev theorem.  Markovs theorem .

Her expectation is:

  13.4.  The generalized Chebyshev theorem.  Markovs theorem ,

and the variance

  13.4.  The generalized Chebyshev theorem.  Markovs theorem .

Apply to value   13.4.  The generalized Chebyshev theorem.  Markovs theorem Chebyshev's inequality:

  13.4.  The generalized Chebyshev theorem.  Markovs theorem ,

or

  13.4.  The generalized Chebyshev theorem.  Markovs theorem . (13.4.2)

Replace the right side of the inequality (13.4.2) each of the quantities   13.4.  The generalized Chebyshev theorem.  Markovs theorem larger value   13.4.  The generalized Chebyshev theorem.  Markovs theorem . Then the inequality will only intensify:

  13.4.  The generalized Chebyshev theorem.  Markovs theorem .

However small   13.4.  The generalized Chebyshev theorem.  Markovs theorem , can choose   13.4.  The generalized Chebyshev theorem.  Markovs theorem so large that inequality holds

  13.4.  The generalized Chebyshev theorem.  Markovs theorem ;

then

  13.4.  The generalized Chebyshev theorem.  Markovs theorem ,

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   13.4.  The generalized Chebyshev theorem.  Markovs theorem and if at   13.4.  The generalized Chebyshev theorem.  Markovs theorem

  13.4.  The generalized Chebyshev theorem.  Markovs theorem ,

the arithmetic mean of the observed values ​​of random variables   13.4.  The generalized Chebyshev theorem.  Markovs theorem converges in probability to the arithmetic mean of their mathematical expectations. Evidence. Consider the value

  13.4.  The generalized Chebyshev theorem.  Markovs theorem .

Obviously

  13.4.  The generalized Chebyshev theorem.  Markovs theorem .

Apply to value   13.4.  The generalized Chebyshev theorem.  Markovs theorem Chebyshev's inequality:

  13.4.  The generalized Chebyshev theorem.  Markovs theorem .

As by the condition of the theorem,   13.4.  The generalized Chebyshev theorem.  Markovs theorem   13.4.  The generalized Chebyshev theorem.  Markovs theorem then at a sufficiently large   13.4.  The generalized Chebyshev theorem.  Markovs theorem

  13.4.  The generalized Chebyshev theorem.  Markovs theorem ,

or, moving to the opposite event,

  13.4.  The generalized Chebyshev theorem.  Markovs 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