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Statistical Inference: A Short Course by Michael J. Panik

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5.2 The Mean, Variance, and Standard Deviation of A Discrete Random Variable

The starting point for determining the mean and variance of a discrete random variable X (or of X's discrete probability distribution) is Table 5.1. Looking first to the mean of X, convention dictates that the mean be termed the expectation of X (or the expected value of X) and denoted as follows:

(5.2) equation

Here E(X) serves as a measure of central location (the center of gravity) of a discrete probability distribution. For instance, given the discrete probability distribution appearing in Table 5.5, let us find and interpret E(X).

Table 5.5 A Discrete Probability Distribution.

X f(X)
10 0.30
40 0.70
1.00

As we shall now see, E(X) is the “long-run” average value of X—it represents the mean outcome we expect to obtain if a random experiment is repeated a large number of times in the long run. In this regard, from Table 5.5, we get the following:

img

that is, if we repeat our random experiment over and over a large number of times, then, in the long run, since 10 occurs 30% of the time and 40 occurs 70% of the time, the average value of X will be 31. So if the long run consists of, say, 100 trials, then the average of 30 tens and 70 fortys would be 31.

Why does E(X) have the form provided by Equation ...

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