In probability theory, concentration inequalities provide mathematical bounds on the probability of a random variable deviating from some value (typically, its expected value). The deviation or other function of the random variable can be thought of as a secondary random variable. The simplest example of the concentration of such a secondary random variable is the CDF of the first random variable which concentrates the probability to unity. If an analytic form of the CDF is available this provides a concentration equality that provides the exact probability of concentration. It is precisely when the CDF is difficult to calculate or even the exact form of the first random variable is unknown that the applicable concentration inequalities provide useful insight.
Another almost universal example of a secondary random variable is the law of large numbers of classical probability theory which states that sums of independent random variables, under mild conditions, concentrate around their expectation with a high probability. Such sums are the most basic examples of random variables concentrated around their mean.
Concentration inequalities can be sorted according to how much information about the random variable is needed in order to use them.[citation needed]