121616949-math.231 - 9.8 217 Probability the square root we...

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9.8 Probability 217 the square root we “correct for” the original square, so that the units of σ are the same as the units of X , and σ measures a “typical distance from μ ” instead of a typical squared distance. The standard deviation is very useful in statistical analysis. EXAMPLE 9.36 We compute the standard deviation of the standard normal distru- bution. The variance is 1 2 π Z -∞ x 2 e - x 2 / 2 dx. To compute the antiderivative, use integration by parts, with u = x and dv = xe - x 2 / 2 dx . This gives Z x 2 e - x 2 / 2 dx = - xe - x 2 / 2 + Z e - x 2 / 2 dx. We cannot do the new integral, but we know its value when the limits are -∞ to , from our discussion of the standard normal distribution. Thus 1 2 π Z -∞ x 2 e - x 2 / 2 dx = - 1 2 π xe - x 2 / 2 fl fl fl fl -∞ + 1 2 π Z -∞ e - x 2 / 2 dx = 0 + 1 2 π 2 π = 1 . The standard deviation is then 1 = 1. Here is a simple example of the use of these ideas. Suppose it is known that, in the
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