Chebyshev's inequality

Chebyshev's inequality bounds the probability that a random variable with finite mean and variance differs from its mean by at least a specified amount. The bound is its variance divided by the squared amount.

Markov's inequality: For a nonnegative random variable, Markov's inequality bounds the probability of exceeding a threshold by its expectation divided by that threshold. Applying it to squared deviations from the mean gives Chebyshev's bound.
Weak law of large numbers: The weak law of large numbers states that sample averages converge in probability to the population mean under suitable conditions. Chebyshev's inequality proves the law for independent, identically distributed variables with finite variance.
Irénée-Jules Bienaymé: Irénée-Jules Bienaymé was a French statistician who studied probability, statistics, and the law of large numbers. He derived an early form of the inequality before Chebyshev's influential treatment.
Chernoff bound: A Chernoff bound limits tail probabilities using exponential moments of a random variable. With stronger distributional information, it often gives exponentially tighter tails than Chebyshev's bound.
Variance: Variance is the expected squared deviation of a random variable from its mean. It supplies the numerator by measuring the total squared deviation.
Sample mean: The sample mean is the arithmetic average of observations drawn from a dataset or random sample. Its variance shrinks with sample size, allowing Chebyshev's inequality to certify concentration.
Pafnuty Chebyshev: Pafnuty Chebyshev was a Russian mathematician known for work in probability, number theory, and approximation. His work established the inequality's central role in proving convergence results.
Hoeffding's inequality: Hoeffding's inequality gives exponential tail bounds for sums of independent bounded random variables. Boundedness and independence yield sharper guarantees than finite variance alone.
Indicator random variable: An indicator random variable equals one when an event occurs and zero otherwise. An event indicator lets Markov's inequality bound the probability of a large squared deviation.
Concentration inequality: A concentration inequality bounds the probability that a random quantity departs substantially from a typical value. Chebyshev's bound is a basic concentration result requiring only finite variance.

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