CHEBYSHOV'S THEOREM: WHAT IT IS, APPLICATIONS AND EXAMPLES - MATHS - 2025

The theorem Chebyshev (Chebyshev or inequality) is one of the most important classical results of the theory of probability. It allows estimating the probability of an event described in terms of a random variable X, by providing us with a bound that does not depend on the distribution of the random variable but on the variance of X.

The theorem is named after the Russian mathematician Pafnuty Chebyshov (also written as Chebychev or Tchebycheff) who, despite not being the first to state the theorem, was the first to give a proof in 1867.

This inequality, or those that due to their characteristics are called Chebyshov's inequality, is used mainly to approximate probabilities by calculating heights.

What does it consist of?

In the study of probability theory, it occurs that if the distribution function of a random variable X is known, its expected value -or mathematical expectation E (X) - and its variance Var (X) can be calculated, as long as such amounts exist. However, the converse is not necessarily true.

That is, knowing E (X) and Var (X) it is not necessarily possible to obtain the distribution function of X, therefore quantities such as P (-X-> k) for some k> 0 are very difficult to obtain. But thanks to Chebyshov's inequality it is possible to estimate the probability of the random variable.

Chebyshov's theorem tells us that if we have a random variable X over a sample space S with a probability function p, and if k> 0, then:

Applications and examples

Among the many applications of Chebyshov's theorem, the following can be mentioned:

Limiting probabilities

This is the most common application and is used to give an upper bound for P (-XE (X) -≥k) where k> 0, only with the variance and the expectation of the random variable X, without knowing the probability function.

Example 1

Suppose that the number of products manufactured in a company during a week is a random variable with an average of 50.

If the variance of one week of production is known to be equal to 25, then what can we say about the probability that this week the production will differ by more than 10 from the mean?

Solution

Applying Chebyshov's inequality we have:

From this we can obtain that the probability that in the production week the number of articles exceeds the average by more than 10 is at most 1/4.

Proof of Limit Theorems

Chebyshov's inequality plays an important role in proving the most important limit theorems. As an example we have the following:

Weak law of large numbers

This law states that given a sequence X1, X2,…, Xn,… of independent random variables with the same average distribution E (Xi) = μ and variance Var (X) = σ ², and a known mean sample of:

Then for k> 0 we have:

Or, equivalently:

Demonstration

Let's first notice the following:

Since X1, X2,…, Xn are independent, it follows that:

Therefore, it is possible to state the following:

Then, using Chebyshov's theorem, we have:

Finally, the theorem results from the fact that the limit on the right is zero as n approaches infinity.

It should be noted that this test was made only for the case in which the variance of Xi exists; that is, it does not diverge. Thus we observe that the theorem is always true if E (Xi) exists.

Chebyshov limit theorem

If X1, X2,…, Xn,… is a sequence of independent random variables such that there exists some C <infinity, such that Var (Xn) ≤ C for all natural n, then for any k> 0:

Demonstration

Since the sequence of variances is uniformly bounded, we have that Var (Sn) ≤ C / n, for all natural n. But we know that:

Making n tend towards infinity, the following results:

Since a probability cannot exceed the value of 1, the desired result is obtained. As a consequence of this theorem, we could mention the particular case of Bernoulli.

If an experiment is repeated n times independently with two possible outcomes (failure and success), where p is the probability of success in each experiment and X is the random variable that represents the number of successes obtained, then for each k> 0 you have to:

Sample size

In terms of variance, the Chebyshov inequality allows us to find a sample size n that is sufficient to guarantee that the probability that -Sn-μ -> = k occurs is as small as desired, which allows an approximation to the average.

Specifically, let X1, X2,… Xn be a sample of independent random variables of size n and suppose that E (Xi) = μ and its variance σ ². Then, by Chebyshov's inequality we have:

Example

Suppose that X1, X2,… Xn are a sample of independent random variables with Bernoulli distribution, such that they take the value 1 with probability p = 0.5.

What must be the size of the sample to be able to guarantee that the probability that the difference between the arithmetic mean Sn and its expected value (exceeding by more than 0.1), is less than or equal to 0.01?

Solution

We have that E (X) = μ = p = 0.5 and that Var (X) = σ ² = p (1-p) = 0.25. By Chebyshov's inequality, for any k> 0 we have:

Now, taking k = 0.1 and δ = 0.01, we have:

In this way it is concluded that a sample size of at least 2,500 is needed to guarantee that the probability of the event -Sn - 0.5 -> = 0.1 is less than 0.01.

Chebyshov-type inequalities

There are several inequalities related to Chebyshov's inequality. One of the best known is the Markov inequality:

In this expression X is a non-negative random variable with k, r> 0.

The Markov inequality can take different forms. For example, let Y be a non-negative random variable (so P (Y> = 0) = 1) and suppose that E (Y) = μ exists. Suppose also that (E (Y)) ^r = μ _r exists for some integer r> 1. So:

Another inequality is Gaussian, which tells us that given a unimodal random variable X with mode at zero, then for k> 0,

References

1. Kai Lai Chung. Elementary Proability Theory with Stochastic Processes. Springer-Verlag New York Inc
2. Kenneth.H. Rosen. Discrete Mathematics and its Applications. SAMCGRAW-HILL / INTERAMERICANA DE ESPAÑA.
3. Paul L. Meyer. Probability and Statistical Applications. SA ALHAMBRA MEXICANA.
4. Seymour Lipschutz Ph.D. 2000 Solved Problems of Discrete Mathematics. McGRAW-HILL.
5. Seymour Lipschutz Ph.D. Theory and Probability Problems. McGRAW-HILL.