Markov's inequality bounds the probability that a non-negative random variable exceeds a given threshold using only its expectation, providing a distribution-free but often loose upper bound; Chebyshev's inequality strengthens this by using the variance to bound the probability of deviation from the mean in either direction, yielding tighter bounds when variance is known. The Chernoff bound provides an exponentially strong tail bound for the sum of mutually independent bounded random variables by applying Markov's inequality to an exponentiated (moment-generating-function-based) transformation of the sum, a technique central to probability theory's study of concentration of measure and tail bounds for sums of independent random variables.
Markov's inequality bounds the probability that a non-negative random variable exceeds a given threshold using only its expectation, providing a distribution-free but often loose upper bound; Chebysh…