Both PMF and CDF provide us with all of the relevant information about a discrete random variable, maybe too much. In descriptive statistics, we use summary measures, such as mean, median, mode, variance, and standard deviation, to get a feeling for some essential features of a distribution, like its location and dispersion. In probability theory, there are corresponding concepts that we start exploring in this and the next section, where we consider concepts corresponding to mean and variance, leaving further developments to Sections 7.4 and 7.5. The single most relevant feature of a distribution is related to its “mean,” a natural measure of location.
DEFINITION 6.5 (Expected value) The expected value of a discrete random variable with PMF pX(x) is given by
where we have used the shorthand notation pi = pX(xi).
This definition allows for an infinite support; in the case of a finite support, the sum will just run up to i = N. Quite often, the notation μ or μX is used to refer to the expected value of a random variable.
Example 6.5 Consider again the random variable whose PMF is shown in Table 6.1. Its expected value is
Note that the expected value of a random variable taking integer values is not necessarily an integer number.
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