What is Chebyshev’s inequality and Examples

In this blog, we would discuss What is Chebyshev’s inequality and Examples. In mathematics, Chebyshev’s inequality (also called the Tchebychev inequality) is a result of probability theory that is useful in statistics. It states that for any random variable X with mean μ and standard deviation σ, the following inequality holds:

 

 

Pr(|X−μ|≥kσ)≤1/k2

 

 

In other words, the probability that the random variable X is more than k standard deviations away from its mean is less than 1/k2. Inequality is often used in statistical applications to get bounds on tail probabilities.

 

 

For example, if X is a normal random variable with mean μ and standard deviation σ, then we can use Chebyshev’s inequality to show that Pr(|X−μ|≥3σ)≤1/9 In other words, the probability that X is more than 3 standard deviations away from its mean is less than 1/9.

 

 

This can be useful in situations where we want to be confident that our data is not too far from the mean. There are other inequalities that give similar results, such as the Markov inequality and the Hoeffding inequality.

 

 

 

What is Chebyshev’s inequality? 

Chebyshev’s inequality is a powerful tool for understanding the distribution of data. It states that for any data set, the proportion of values that lie within k standard deviations of the mean is at least 1 – 1/k2. In other words, if you take any data set and plot it on a graph, at least 1 – 1/k2 of the data will lie within k standard deviations of the mean. This inequality is named after Pafnuty Chebyshev, who first proved it in 1867.

 

 

It is a generalization of the empirical rule, which states that for a data set with a normal distribution, 68% of the data lies within 1 standard deviation of the mean, 95% of the data lies within 2 standard deviations of the mean, and 99.7% of the data lies within 3 standard deviations of the mean. Chebyshev’s inequality is much more general than the empirical rule. It applies to any data set, regardless of its distribution. It is a useful tool for understanding the distribution of data and can be used to make predictions about how likely it is for a value to lie within a certain range.

 

 

 

Examples of Chebyshev’s inequality

Let X be a random variable with mean μ and standard deviation σ. Then, for any real number a > 0, Pr(|X – μ| ≥ a) ≤ 1/a².

 

 

In other words, the probability that X is more than a standard deviation away from the mean is always less than 1/a². The inequality can be rewritten in a form that is sometimes more useful:

 

 

Pr(X ≥ μ + a) ≤ 1/a² Pr(X ≤ μ – a) ≤ 1/a²

 

 

This form states that the probability that X is more than a certain number away from the mean is always less than 1/a². The inequality is useful because it gives a bound on how “unlikely” it is for a random variable to take a value that is far away from the mean.

 

 

Now we will look into Examples of Chebyshev’s inequality

 

 

For example, if X is a normally distributed random variable with mean μ = 0 and standard deviation σ = 1, then we can use Chebyshev’s inequality to say that: Pr(|X| ≥ 3) ≤ 1/9 In other words, the probability that X is more than 3 standard deviations away from the mean is less than 1/9. This is a very small probability, so we can say that it is very unlikely for X to take a value that is more than 3 standard deviations away from the mean.

 

 

Similarly, we can use Chebyshev’s inequality to say that: Pr(|X| ≥ 4) ≤ 1/16 In other words, the probability that X is more than 4 standard deviations away from the mean is less than 1/16. This is an even smaller probability, so we can say that it is even more unlikely for X to take a value that is more than 4 standard deviations away from the mean. We can keep using Chebyshev’s inequality to get bounds on how unlikely it is for X to take a value that is more than n standard deviations away from the mean, for any positive integer n.

 

For example

 

 Pr(|X| ≥ 5) ≤ 1/25

Pr(|X| ≥ 6) ≤ 1/36

Pr(|X| ≥ 7) ≤ 1/49

…

Pr(|X| ≥ n) ≤ 1/n²

 

 

In general, we can say that the probability that X is more than n standard deviations away from the mean is less than 1/n². It is important to note that Chebyshev’s inequality is a statement about probabilities, so it is not always possible to say that X will not take a value that is more than n standard deviations away from the mean. It is possible for X to take any value, no matter how unlikely it may be.

 

 

However, we can use Chebyshev’s inequality to say that the probability of X taking a value that is more than n standard deviations away from the mean is very small.

 

 

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