Solving Probability of Random Variables Using Chebyshev's Inequality

In this article, we will explore a classic problem in probability theory and statistics: finding the probability that a random variable deviates from its mean by a certain amount. Specifically, we will use Chebyshev's Inequality to determine P[X 2] for a given random variable X.

Assumptions and Background

To solve this problem, we'll make three key assumptions. First, P represents a probability function. Second, X is a random variable with a known mean μ and variance σ2. Third, we will assume that the mean μ of X is 0 for simplicity. Given these assumptions, we can calculate the probability that X exceeds a specific threshold using Chebyshev's Inequality.

Chebyshev's Inequality

Chebyshev's Inequality is a powerful tool that provides an upper bound on the probability that a random variable deviates from its mean by a certain amount. Mathematically, it states:

[P[|X - mu| gamma] leq frac{sigma^2}{gamma^2}]

Here, γ is the deviation threshold. In our case, we set γ 2. Let's apply this to our problem.

Application to our Problem

Substituting γ 2 into Chebyshev's Inequality, we get:

[P[|X - mu| 2] leq frac{sigma^2}{4}]

Since we assumed μ 0, this simplifies to:

[P[X 2] leq frac{sigma^2}{4}]

Standard Normal and Uniformly Distributed Random Variables

To illustrate the practical implications of Chebyshev's Inequality, let's consider the case of a standard normal and a uniformly distributed random variable. The probabilities P[X 2] and the corresponding Chebyshev's bounds are shown in the figure below:

Proof of Chebyshev's Inequality

The proof of Chebyshev's Inequality can be derived from the definition of variance:

[text{Var}[X] int_{mathbb{R}} (x - mu)^2 p_X(x) , dx]

Breaking this into two regions, we get:

[text{Var}[X] int_{{|x - mu| gamma}} (x - mu)^2 p_X(x) , dx int_{{|x - mu| leq gamma}} (x - mu)^2 p_X(x) , dx]

Since [(x - mu)^2 gamma^2] in the first integral, we have:

[text{Var}[X] geq int_{{|x - mu| gamma}} (x - mu)^2 p_X(x) , dx geq int_{{|x - mu| gamma}} gamma^2 p_X(x) , dx gamma^2 P[|X - mu| gamma]]

Thus, we obtain:

[P[|X - mu| gamma] leq frac{text{Var}[X]}{gamma^2}]

Conclusion

In this article, we have demonstrated how to use Chebyshev's Inequality to find an upper bound on the probability that a random variable deviates from its mean. This inequality is particularly useful when we have limited information about the distribution of the random variable. By applying Chebyshev's Inequality, we can make valuable probabilistic statements about the behavior of the random variable.

For further reading and a deeper understanding, you may want to explore the following resources:

Additional Information on Chebyshev's Inequality Applications of Chebyshev's Inequality in Real-World Scenarios Problems and Exercises on Chebyshev's Inequality