Ralph Vince - Portfolio Mathematics

ch02 JWBK035-Vince February 12, 2007 6:50 Char Count= 0

Probability Distributions 65

FIGURE 2.12 The area between−1and+2 standard units

the probability of not exceeding−1 standard unit and the probability of

exceeding 2 standard units:

= 1 −(. 022750052 +.1586552595)

= 1 −. 1814053117

=. 8185946883

With the basic mathematical tools regarding the Normal Distribution

thus far covered in this chapter, you can now use your powers of reasoning

to figure any probabilities of occurrence for Normally distributed random

variables.

Further Derivatives of the Normal

Sometimes you may want to know the second derivative of the N(Z) func-

tion. Since the N(Z) function gives us the area under the curve at Z, and the

N′(Z) function gives us the height of the curve itself at Z, then the N′′(Z)

function gives us theinstantaneous slopeof the curve at a given Z:

N′′(Z)=−Z/ 2. 506628274 ∗EXP(−(Z^2 )/2) (2.23)

where: EXP( )=The exponential function.