Question

What is Chebyshev’s inequality?

Answer 1

Hint: We need to explain the concept of Chebyshev’s inequality by using the concept of an example. Chebyshev’s inequality is a concept used for random variables to determine the maximum number of extreme values. Its equation can be represented by $\Pr \left( \left| X-\mu \right|\ge k\sigma \right)\le \dfrac{1}{{{k}^{2}}}.$

Complete step by step solution:
In order to answer this question, let us first explain the concept of Chebyshev’s inequality. It is an important concept in probability used to determine the extreme values. We can write its equation as follows,
$\Rightarrow \Pr \left( \left| X-\mu \right|\ge k\sigma \right)\le \dfrac{1}{{{k}^{2}}}$
Here, X is a random variable having a mean given by $\mu ,$ and a finite variance ${{\sigma }^{2}}.$ k is a positive real number such that $k\in {{R}^{+}}$ which represents a certain distance away from the mean expressed in terms of standard deviation. Let us consider the example of calculating the event of tossing a coin 10 times in which we are required to calculate the probability of getting the number of tails greater than 7 or less than 3. The boundary value can be found using Chebyshev's inequality.
Let X be the number of tails. The expected value is given by the product of the number of tries and the probability of getting a tail which is 0.5
$\Rightarrow \mu =n\times p=10\times 0.5=5$
The variance can be calculated as,
$\Rightarrow {{\sigma }^{2}}=n\times p\times \left( 1-p \right)=10\times 0.5\times 0.5=2.5$
Standard deviation can be calculated as
$\Rightarrow \sqrt{{{\sigma }^{2}}}=\sqrt{2.5}=1.5811$
This is the same as the value k. Hence, using Chebyshev’s inequality for more than 7 tails and less than 3 tails can be given by,
$\Rightarrow \Pr \left( X<3\bigcup X>7 \right)=\Pr \left( \left| X-\mu \right|\ge k\sigma \right)\le \dfrac{1}{{{k}^{2}}}=\dfrac{1}{{{1.5811}^{2}}}$
This can be simplified to,
$\Rightarrow \Pr \left( X<3\bigcup X>7 \right)=\Pr \left( \left| X-\mu \right|\ge k\sigma \right)\le \dfrac{1}{2.4998}=0.4$
Hence, the maximum probability for the case of obtaining more than 7 tails and less than 3 tails is given by 0.4.
Hence, we have explained the concept of Chebyshev’s inequality.

Note: We need to note that Chebyshev's inequality is an important concept used for many mathematical problems. It can be used to determine the outliers of a group of clustered data. It can also determine the characteristics of probability distributions.