For any non-negative random variable X and a > 0, P(X ≥ a) ≤ E[X]/a. A. Markov 1884. Immediate from E[X] ≥ E[X·1_{X≥a}] ≥ a·P(X ≥ a). Parent of Chebyshev (via (Y−μ)²) and Chernoff (via e^{λY}). Sharp at two-point distribution on {0, a}.
For any non-negative random variable X and a > 0, P(X ≥ a) ≤ E[X]/a. A. Markov 1884. Immediate from E[X] ≥ E[X·1_{X≥a}] ≥ a·P(X ≥ a). Parent of Chebyshev (via (Y−μ)²) and Chernoff (via e^{λY}). Sharp at two-point distribution on {0, a}.