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In Chapter 2 we found that the sample variance was one useful measure of the dispersion of a set of numerical observations. The sample variance is the average of the squared discrepan- cies of the observations from their mean. We use this same idea to measure dispersion in the probability distribution of a random variable. We define the variance of a random variable as the weighted average of the squares of its possible deviations, (x - ), from the mean; the weight associated with (x m.) is the probability that the random variable takes the valuex. The variance can then be viewed as the average value that will be taken by the function (X Mover a very large number of repeated trials, as defined by Equation 4.5. Variance and Standard Deviation of a Discrete Random Variable Let X be a discrete random variable. The expectation of the squared deviations about the mean, (X - M), is called the variance, denoted as o and given by in = E[(x - x)2) = S(x-4)*P(x) (4.5) The variance of a discrete random variable X can also be expressed as o? = E(X?1 - ? - SIP(x) - 14.6) The standard deviation, or, is the positive square root of the variance. 4.3. Properties of Discrete Random Variables 133 t10 In some practical applications the alternative, but equivalent, formula for the vari- ance is preferable for computational purposes. That alternative formula is defined by Equation 4.6, which can be verified algebraically (see the chapter appendix), The concept of variance can be very useful in comparing the dispersions of probabil- ity distributions. Consider, for example, viewing as a random variable the daily return over a year on an investment. Two investments may have the same expected returns but will still differ in an important way if the variances of these returns are substantially dif- ferent. A higher variance indicates that returns substantially different from the mean are more likely than if the variance of returns amount is small. In this context, then, variance of the return can be associated with the concept of the risk of an investment--the higher the variance, the greater the risk Taking the square root of the variance to obtain the standard deviation yields a quan- tity in the original units of measurement, as noted in Chapter 2: Example 4.4 Expected Value and Variance of Automobile Sales (Expected Value and Variance) In Example 4.2 Olaf Motors, Inc., determined that the number of Prius cars sold daily could vary from 0 to 5, with the probabilities given in Table 4.2. Find the expected value and variance for this probability distribution