ABSTRACT
In this chapter, the authors introduce several important inequalities in probability. These inequalities will often allow to narrow down the range of possible values for the exact answer, that is, to determine an upper bound and/or lower bound. Cauchy-Schwarz also allows to deduce the existence of a joint moment generating function (MGF) from the existence of marginal MGFs; this is another example of the benefit of being able to bound a joint distribution quantity by marginal distribution quantities. The Cauchy-Schwarz inequality is one of the most famous inequalities in all of mathematics. The inequalities provide bounds on the probability of a random variable taking on an “extreme” value in the right or left tail of a distribution. The authors explain limit theorems to study a couple of important named distributions in statistics.
