1 Markov and Chebyshev

Theorem 1. Markov: Let X be a random variable assuming only non-negative values. Then for all positive real numbers:

P (X ≥ t) ≤ E(X)/t

Equivalently

P (X ≥ kE(X)) ≤ 1/k

Proof by contradiction.

For a random variable X with expectation μ_{x}, its variance σ_{x}^{2} is defined to be E(X − μ_{x}^{2})