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

13.2. Chebyshev Inequality

Lecture



As a lemma, which is necessary for the proof of theorems belonging to the group of the “law of large numbers,” we will prove one very general inequality, known as the Chebyshev inequality.

Let there be a random variable   13.2.  Chebyshev Inequality with mathematical expectation   13.2.  Chebyshev Inequality and variance   13.2.  Chebyshev Inequality . Chebyshev's inequality states that, whatever the positive number   13.2.  Chebyshev Inequality the probability that the magnitude   13.2.  Chebyshev Inequality deviate from their expectation by no less than   13.2.  Chebyshev Inequality , bounded above by value   13.2.  Chebyshev Inequality :

  13.2.  Chebyshev Inequality . (13.2.1)

Evidence. 1. Let the value   13.2.  Chebyshev Inequality discontinuous, with a number of distribution

  13.2.  Chebyshev Inequality

  13.2.  Chebyshev Inequality

  13.2.  Chebyshev Inequality

  13.2.  Chebyshev Inequality

  13.2.  Chebyshev Inequality

  13.2.  Chebyshev Inequality

  13.2.  Chebyshev Inequality

  13.2.  Chebyshev Inequality

  13.2.  Chebyshev Inequality

  13.2.  Chebyshev Inequality

We depict the possible values ​​of   13.2.  Chebyshev Inequality and its expectation   13.2.  Chebyshev Inequality in the form of points on the number axis   13.2.  Chebyshev Inequality (fig. 13.2.1).

  13.2.  Chebyshev Inequality

Fig. 13.2.1.

Let's set some value   13.2.  Chebyshev Inequality and calculate the probability that the magnitude   13.2.  Chebyshev Inequality deviate from their expectation by no less than   13.2.  Chebyshev Inequality :

  13.2.  Chebyshev Inequality . (13.2.2)

To do this, set aside from the point   13.2.  Chebyshev Inequality right and left along the length segment   13.2.  Chebyshev Inequality ; get the segment   13.2.  Chebyshev Inequality . Probability (13.2.2) is nothing but the probability that a random point   13.2.  Chebyshev Inequality will not fall inside the segment   13.2.  Chebyshev Inequality , and outside it:

  13.2.  Chebyshev Inequality .

In order to find this probability, you need to sum up the probabilities of all those values   13.2.  Chebyshev Inequality which lie outside the segment   13.2.  Chebyshev Inequality . We will write this as follows:

  13.2.  Chebyshev Inequality (13.2.3)

where is the record   13.2.  Chebyshev Inequality under the sum sign means that the summation applies to all those values   13.2.  Chebyshev Inequality for which points   13.2.  Chebyshev Inequality , lie outside the segment   13.2.  Chebyshev Inequality .

On the other hand, we write the expression for the variance of   13.2.  Chebyshev Inequality . By definition:

  13.2.  Chebyshev Inequality . (13.2.4)

Since all members of the sum (13.2.4) are non-negative, it can only decrease if we do not extend it to all values   13.2.  Chebyshev Inequality , but only on some, in particular on those that lie outside the segment   13.2.  Chebyshev Inequality :

  13.2.  Chebyshev Inequality . (13.2.5)

Replace the expression with the sum sign   13.2.  Chebyshev Inequality through   13.2.  Chebyshev Inequality . Since for all members of the sum   13.2.  Chebyshev Inequality then the amount of such a replacement can also only decrease; means

  13.2.  Chebyshev Inequality . (13.2.6)

But according to formula (13.2.3), the sum on the right side of (13.2.6) is nothing more than the probability of a random point falling outside the segment   13.2.  Chebyshev Inequality ; Consequently,

  13.2.  Chebyshev Inequality ,

whence the derived inequality directly follows.

2. In the case where the value   13.2.  Chebyshev Inequality is continuous, the proof is carried out in a similar way with the replacement of probabilities   13.2.  Chebyshev Inequality element of probability, and finite sums - integrals. Really,

  13.2.  Chebyshev Inequality . (13.2.7)

Where   13.2.  Chebyshev Inequality - distribution density   13.2.  Chebyshev Inequality . Next, we have:

  13.2.  Chebyshev Inequality

  13.2.  Chebyshev Inequality ,

where is the sign   13.2.  Chebyshev Inequality under the integral means that the integration extends to the outer part of the segment   13.2.  Chebyshev Inequality .

Replacing   13.2.  Chebyshev Inequality under the sign of the integral through   13.2.  Chebyshev Inequality , we get:

  13.2.  Chebyshev Inequality ,

whence Chebyshev's inequality for continuous quantities follows.

Example. Given a random variable   13.2.  Chebyshev Inequality with mathematical expectation   13.2.  Chebyshev Inequality and variance   13.2.  Chebyshev Inequality . Estimate the top probability of the magnitude   13.2.  Chebyshev Inequality deviate from their expectation by no less than   13.2.  Chebyshev Inequality .

Decision. Believing in Chebyshev's inequality   13.2.  Chebyshev Inequality , we have:

  13.2.  Chebyshev Inequality ,

i.e. the probability that the deviation of a random variable from its expectation will go beyond the three mean square deviations cannot be greater than   13.2.  Chebyshev Inequality .

Note. Chebyshev inequality gives only the upper limit of the probability of a given deviation. Above this limit, probability cannot be under any distribution law. In practice, in most cases the probability that the magnitude   13.2.  Chebyshev Inequality go outside the lot   13.2.  Chebyshev Inequality , significantly less   13.2.  Chebyshev Inequality . For example, for a normal law, this probability is approximately equal to 0.003. In practice, most often we deal with random variables, the values ​​of which only very rarely go beyond   13.2.  Chebyshev Inequality . If the distribution law of a random variable is unknown, but only   13.2.  Chebyshev Inequality and   13.2.  Chebyshev Inequality , in practice, the segment is usually considered   13.2.  Chebyshev Inequality a section of practically possible values ​​of a random variable (the so-called “three sigma rule”).


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