Chebyshev Inequality

Worked Example

Let's calculate the exact tail probability and the Chebyshev bound for a fair six-sided die roll. The outcomes are the set {1, 2, 3, 4, 5, 6}, each with probability 1/6.

• Mean (μ): (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5
• Variance (σ²): The expected value of (X - 3.5)². Calculating this yields exactly 35/12 ≈ 2.916667.

Suppose we want to bound the probability that the roll deviates from the mean by at least t = 2.

1. Calculate the actual tail probability:The condition |X - 3.5| ≥ 2 is satisfied when X - 3.5 ≥ 2 (meaning X ≥ 5.5) or X - 3.5 ≤ -2 (meaning X ≤ 1.5). Since X must be an integer, this means X = 1 or X = 6. There are 2 such outcomes, so the true probability is 2/6 = 1/3 ≈ 0.333333.
2. Calculate the Chebyshev Bound:Using the formula σ² / t², we compute (35/12) / (2²) = (35/12) / 4 = 35/48 ≈ 0.729167.
3. Compare:The true tail probability (0.333333) is indeed less than or equal to the Chebyshev bound (0.729167). The bound is loose in this case because a uniform die roll doesn't concentrate all its variance at the extremes.

Experiment

You can use the Chebyshev Inequality interactive module to explore when this bound is sharp and when it is loose.

1. Select the Three-point distribution preset. This distribution has atoms at -2, 0, and 2 with weights 1, 2, and 1, respectively. The mean is 0 and the variance is 2.
2. Set the Radius t to 1. The inclusive tail |X| ≥ 1 includes the atoms -2 and 2, which contain 50% (0.5) of the probability mass. The Chebyshev bound is min(1, 2/1²) = 1.0.
3. Increase the Radius t to 2. The inclusive tail |X| ≥ 2 still only includes the atoms -2 and 2, so the actual probability remains 0.5. However, the Chebyshev bound is now min(1, 2/2²) = 0.5. Here, the bound perfectly matches the exact probability. This is a sharp example!
4. Change the preset to Two-point, placing probability mass at -1 and 1. At Radius t = 1, the true tail probability is 1.0, and the Chebyshev bound is exactly 1.0.
5. Decrease the radius to t = 0 for the Three-point distribution. The actual probability of deviating by at least 0 is 1.0, but the Chebyshev bound is undefined because division by zero is not permitted. The interface clearly marks it as Unavailable for radius0.

Limitations and Boundaries

There are a few important limitations to keep in mind when interpreting the results of the interactive module and the Chebyshev inequality generally.

• Finite Support: This module explores exactly three fixed discrete presets. In reality, the Chebyshev inequality applies equally to continuous distributions, provided their variance is finite.
• Zero Variance: If a distribution has zero variance (e.g., the Point mass preset), it perfectly equals its mean. For any positive radius, the true probability of deviation is 0, and the Chebyshev bound correctly returns 0.
• Radius Zero: While the mathematical inclusive event |X - μ| ≥ 0 always has a probability of 1, the formula σ² / t² requires division by t. Therefore, the numerical bound is undefined at t = 0.
• Trivial Bounds: For very small positive values of t, σ² / t² can be significantly larger than 1. Because probabilities cannot exceed 1, the useful bound is effectively min(1, σ² / t²).

References

Further reading on the derivation and interpretation of the Chebyshev Inequality.

• Introduction to Probability, Statistics, and Random Processes - 6.2.2 Markov and Chebyshev Inequalities
  This textbook section by Pishro-Nik thoroughly explains the inclusive deviation event and the absolute requirement for a strictly positive radius condition.