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Table Of Contents
Mathematics of Machine Learning
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In this chapter, we have learned about the concept of the expected value. Mathematically speaking, the expected value is defined by
for discrete random variables and
for continuous ones. Although these formulas involve possibly infinite sums and integrals, the underlying meaning is simple: 𝔼[X] represents the average outcome of X, weighted by the underlying probability distribution.
According to the law of large numbers, the expected value also describes a long-term average: if the independent and identically distributed random variables X1,X2,… describe the outcomes of a repeated experiment — say, betting a hand in poker — then the sample average converges to the joint expected value, that is,
holds with probability 1. In a sense, the law of large numbers allows you to glimpse into the future and see what happens if you make the same choice. In the case of poker, if you only make bets with a positive expected value, you’ll win...