Key Inequalities

Markov, Chebyshev, and Jensen — bounding probabilities without the full distribution.

Sometimes you don't know the full distribution of a random variable, but you still need to bound how often it strays. Three inequalities give you a lot of mileage with very little information.

Markov's inequality: for any non-negative $X$ and $a > 0$ ,

P(X \ge a) \le \frac{E[X]}{a}

It says the tail of a non-negative random variable can't be too heavy if the mean is small.

Chebyshev's inequality applies to any variable with mean $\mu$ and variance $\sigma^2$ :

P(|X - \mu| \ge k\sigma) \le \frac{1}{k^2}

So at most $1/k^2$ of the probability mass sits more than $k$ standard deviations from the mean — for any distribution.

Jensen's inequality bounds the expectation of a function of $X$ . For a convex function $\varphi$ ,

\varphi(E[X]) \le E[\varphi(X)]

with the inequality reversed for concave $\varphi$ . Jensen is the secret behind option pricing: option payoffs are convex, so $E[\text{payoff}]$ exceeds the payoff at the expected price — that's where time value comes from.