Normal Distribution, Probability, and Modern Financial Theory
The essence of this chapter is to acquaint you with the basics, plus a tad more on what could be construed to be the core of many inferences in probability theory, the normal distribution. We do not intend to dazzle or overwhelm with mathematics those of you less inclined to enjoy equation after equation. Conversely, any mathematician or probability expert is only asked to read this chapter for the relationship it has to the theory presented in the next chapter.
The equation for the standard form of the normal distribution is:
Any statistics textbook will tell you that the variable z is normally distributed with a mean, μ (Greek letter “mu”) of 0 and a standard deviation, σ (Greek letter “sigma”) of 1. Thus, mathematicians can state that the variable z is normally distributed and, using integration formulae, we can further infer that the areas under the curve between z equals –1 and 1, –2 and 2, and –3 and 3 are equal. The area included between z equals –1 and 1 encompasses 68.27 percent of the area under the curve. Between z equals –2 and 2, the area equals 95.45 percent, and between z equals –3 and 3, the area equals a whopping 99.73 percent! In theory, then, virtually 100 percent of all the area lies between z equals –4 and 4. This equation is graphed in Figure 8.1.