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62 HANDBOOK OF PORTFOLIO MATHEMATICS
FIGURE 2.9 Doing away with the 1−and−Z provision in Equation (2.21)“If Z<0 then N(Z)=1 – N(Z)”). Therefore, the second to last line in the
last computation would be changed from= 1 −. 02275005216to simply. 02275005216
We would therefore say that there is about a 2.275% chance that an
event in a Normally distributed random process would equal or exceed+ 2
standard units. This is shown in Figure 2.9.
Thus far we have looked at areas under the curve (probabilities) where
we are dealing only with what are known as “one-tailed” probabilities. That
is to say we have thus far looked to solve such questions as, “What are the
probabilities of an event’s being less (more) than such-and-such standard
units from the mean?” Suppose now we were to pose the question as, “What
are the probabilities of an event’s beingwithinso many standard units of
the mean?” In other words, we wish to find out what the “2-tailed” proba-
bilities are.
Consider Figure 2.10. This represents the probabilities of being within
2 standard units of the mean. Unlike Figure 2.8, this probability compu-
tation does not include the extreme left tail area, the area of less than− 2
standard units. To calculate the probability of being within Z standard units
of the mean, you must first calculate the one-tailed probability of the abso-
lute value of Z with Equation (2.21). This will be your input to the next