Probability Inequalities: Chebyshev, Markov, and Jensen

Markov's Inequality

If X >= 0 and E[X] exists, then P(X >= a) <= E[X] / a for any a > 0. This is the simplest probability bound and requires only that X is non-negative with a finite mean. While the bound is often loose, it serves as a starting point for tighter inequalities.

Example: If expected claim cost is $500, then P(claim >= $2000) <= 500/2000 = 0.25. The actual probability is likely much smaller, but Markov provides a guaranteed upper bound.

Chebyshev's Inequality

For any random variable X with mean mu and finite variance sigma^2, P(|X - mu| >= k*sigma) <= 1/k^2 for k > 0. Equivalently, P(|X - mu| >= a) <= sigma^2 / a^2. Chebyshev is tighter than Markov because it uses both mean and variance information.

Example: If claims have mean 500 and standard deviation 100, P(|X - 500| >= 300) <= (100/300)^2 = 1/9. So at least 8/9 of claims fall between 200 and 800. Chebyshev is used to prove the Weak Law of Large Numbers.

Jensen's Inequality

If g is a convex function, then E[g(X)] >= g(E[X]). If g is concave, the inequality reverses. Common applications on Exam P: since x^2 is convex, E[X^2] >= (E[X])^2, confirming Var(X) >= 0. Since ln(x) is concave, E[ln(X)] <= ln(E[X]). Jensen's inequality is useful for bounding expectations of transformed random variables when the exact distribution is unknown.