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Handbook of Probability by Ciprian A. Tudor, Ionut Florescu

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Chapter Thirteen

Appendix A: Integration Theory. General Expectations

In this appendix we formalize the theory of calculating expectations. We learned about random variables and their distribution. This distribution completely characterizes a random variable. But in general, distributions are very complex functions. The human brain cannot comprehend such things easily. So the human brain wants to talk about one typical value. For example, one can give a distribution for the random variable representing player salaries in the NBA. Here the variability (probability space) is represented by the specific player chosen. However, suppose we simply want to know the typical salary in the NBA. We probably contemplate a career in sports and want to find out if as an athlete we should go for basketball or baseball. Thus, a single number corresponding to each of these distributions would be much easier to compare. In general, if the distribution is discrete or continuous, then calculating expectations by means we have seen already (summation or integration using probability mass function ir probability density function) will suffice. However, suppose the distribution is more complex so that its c.d.f. is not continuous. For example, suppose that the salary of said athlete depends on whether or not he/she gets injured, whether or not is a male or female, the severity of the injury, and so on. To be able to calculate expectations in this more realistic case, we need to introduce a more complex ...

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