Paley–Zygmund Lower Bound

The Paley–Zygmund inequality is a fundamental result in probability theory that guarantees a lower bound on the probability that a nonnegative random variable is bounded away from zero. It states that for any nonnegative random variable X with finite variance, and for any constant 0 ≤ θ ≤ 1, the probability that X exceeds a fraction θ of its expected value is bounded below by a term involving its first and second moments.

Experiment

This experiment demonstrates how to use the interactive Paley-Zygmund Lower Bound module to investigate the relationship between the actual strict tail probability and its lower bound.

1. Open the Paley–Zygmund Lower Bound module.
2. Set Trials to 4 and Success p to 0.5.
3. Set Threshold theta to 0.5. Observe the threshold marked on the graph.
4. The exact tail probability is 0.6875, and the Paley-Zygmund lower bound is 0.2. Note the bound slack of 0.4875.
5. Change Threshold theta to 1. The lower bound becomes 0, showing that for θ=1 the bound evaluates trivially.