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54 HANDBOOK OF PORTFOLIO MATHEMATICS
well take their sums). Now, we replace those five elements back into the
population, and we take another sample and calculate the sample mean.
If we keep on repeating this process, we will see that the sample means
are Normally distributed, even though the population from which they are
drawn is uniformly distributed.
Furthermore, this is trueregardless of how the population is dis-
tributed! The Central Limit Theorem allows us to treat the distribution of
sample means as being Normal without having to know the distribution
of the population. This is an enormously convenient fact for many areas of
study.
If the population itself happens to be Normally distributed, then the
distribution of sample means will be exactly (not approximately) Normal.
This is true because how quickly the distribution of the sample means
approaches the Normal, as N increases, is a function of how close the
population is to Normal. As a general rule of thumb, if a population has
aunimodal distribution—any type of distribution where there is a con-
centration of frequency around a single mode, and diminishing frequencies
on either side of the mode (i.e., it is convex)—or is uniformly distributed,
using a value of 20 for N is considered sufficient, and a value of 10 for N
is consideredprobablysufficient. However, if the population is distributed
according to the Exponential Distribution (Figure 2.6), then it may be nec-
essary to use an N of 100 or so.
The Central Limit Theorem, this amazingly simple and beautiful fact,
validates the importance of the Normal Distribution.Working with the Normal Distribution
In using the Normal Distribution, we most frequently want to find the per-
centage of area under the curve at a given point along the curve. In the
parlance of calculus this would be called the integral of the function for
the curve itself. Likewise, we could call the function for the curve itself
the derivative of the function for the area under the curve. Derivatives are
often noted with a prime after the variable for the function. Therefore, if
we have a function, N(X), that represents the percentage of area under the
curve at a given point, X, we can say that the derivative of this function,
N′(X) (called N prime of X), is the function for the curve itself at point X.
We will begin with the formula for the curve itself, N′(X). This function
is represented as:N′(X)= 1 /(S∗