Ralph Vince - Portfolio Mathematics

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

Probability Distributions 63

FIGURE 2.10 A two-tailed probability of an event’s being+or−2sigma

Equation, (2.22), which gives us the two-tailed probabilities (i.e., the prob-

abilities of being within ABS(Z) standard units of the mean):

Two-tailed probability= 1 −((1−N(ABS(Z)))∗2) (2.22)

If we are considering what our probabilities of occurrence within 2

standard deviations are (Z=2), then from Equation (2.21) we know that

N(2)=.9772499478, and using this as input to Equation (2.22):

Two-tailed probability= 1 −((1−.9772499478)∗2)

= 1 −(. 02275005216 ∗2)

= 1 −. 04550010432

=. 9544998957

Thus, we can state from this equation that the probability of an event in a

Normally distributed random process falling within 2 standard units of the

mean is about 95.45%.

Just as with Equation (2.21), we can eliminate the leading 1−in Equa-

tion (2.22) to obtain (1 – N(ABS(Z))) * 2, which represents the probabilities

of an event’s falling outside of ABS(Z) standard units of the mean. This is

depicted in Figure 2.11. For the example where Z=2, we can state that the