Verify that the parameter \(\sigma^{2}\) in the expression for the normal density on page 140 is, in fact, its variance.

5.2 The Normal Distribution Among the special probability densities we study in this chapter, the normal prob- ability density, usually referred to simply as the normal distribution, is by far the most important. It was studied first in the eighteenth century when scientists ob- served an astonishing degree of regularity in errors of measurement. They found that the patterns (distributions) they observed were closely approximated by a continu- ous distribution, which they referred to as the "normal curve of errors" and attributed to the laws of chance. The equation of the normal probability density, whose graph (shaped like the cross section of a bell) is shown in Figure 5.3, is