An Introduction to Probability and Statistical Inference, 2nd Edition

3.4 Median and Mode of a Random Variable

Although the mean of a r.v. X does specify the location of the center of the distribution of X (in the sense of the center of gravity), sometimes this is not what we actually wish to know. A case in point is the distribution of yearly income in a community (e.g., in a state or in a country). For the sake of illustration, consider the following (rather) extreme example. A community consisting of 10 households comprises 1 household with yearly income $500,000 and 9 households with respective yearly incomes x_i = $20,000 + $1000(i − 2), i = 2,…, 10. Defining the r.v. X to take the values x = $500,000 and x_i, i = 2,…, 10 with respective probabilities 0.10, we obtain: EX = $71,600. Thus, the average yearly ...