Probability Inequalities

For a set A, let m A = inf{g(t); t ∈ A} for a positive function g. Then The Chebyshev inequality occurs by taking g to be function increasing on the support of X and A = [x, ∞), then m A = g(x), Eg(X) ≥ g(x)P {X > x} or P {X > x} ≤ Eg(X) g(x). This can be seen graphically in Figure 1 for the case g(x) = x. The area of the rectangle xP {X > x} is less than EX, the area above the graph of the cumulative distribution function and below the line y = 1.