Chebyshev Inequality

The Chebyshev Inequality provides a universal upper bound on the probability that a random variable deviates from its mean by at least a specified positive radius. Unlike bounds that require knowing the exact shape of a distribution (such as a normal or binomial distribution), the Chebyshev Inequality relies only on the mean and variance.

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.