A parameter that characterizes the center and location of a pdf is useful as a simple single descriptor of a random variable, as was used in Chapter 3 when we described the properties of several distributions. There are three such measures that generally give different values, although they coincide for some distributions.

Definition: Mean The mean of random variable X is its expectation:

(5.185) Numbered Display Equation

The mean of a random variable can be viewed as that value obtained by averaging the outcomes over a very large number of experiments. It can also be viewed functionally as the area of FX(x) weighted by the odd function g(x) = x which places increasing emphasis on larger values (positive and negative) of x.

Definition: Median The median me of continuous random variable X is the midpoint of the cdf:

(5.186) Numbered Display Equation

which is equivalent to . For a discrete random variable, this last expression is modified to two conditions:

(5.187) Numbered Display Equation

where the directions of the inequalities relative to 1/2 are the same. Substituting the pdfs yields



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