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Inequalities

Concentration inequalities control the likelihood of a random variable taking on large values. Markov's inequality is among the best-known and simplest to prove: for a nonnegative random variable X and any positive number a, it states that P (X ≥ a) ≤ E a. {\displaystyle \operatorname {P} (X\geq a)\leq {\frac {\operatorname {E} }{a}}.}

If X is any random variable with finite expectation, then Markov's inequality may be applied to the random variable | X −E| to obtain Chebyshev's inequality P (| X − E | ≥ a) ≤ Var a 2, {\displaystyle \operatorname {P} (|X-{\text{E}}|\geq a)\leq {\frac {\operatorname {Var} }{a^{2}}},} where Var is the variance. These inequalities are significant for their nearly complete lack of conditional assumptions. For example, for any random variable with finite expectation, the Chebyshev inequality implies that there is at least a 75% probability of an outcome being within two standard deviations of the expected value. However, in special cases the Markov and Chebyshev inequalities often give much weaker information than is otherwise available. For example, in the case of an unweighted dice, Chebyshev's inequality says that odds of rolling between 1 and 6 is at least 53%; in reality, the odds are of course 100%. The Kolmogorov inequality extends the Chebyshev inequality to the context of sums of random variables.

The following three inequalities are of fundamental importance in the field of mathematical analysis and its applications to probability theory.

The Hölder and Minkowski inequalities can be extended to general measure spaces, and are often given in that context. By contrast, the Jensen inequality is special to the case of probability spaces.