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52 HANDBOOK OF PORTFOLIO MATHEMATICS
Finally, it should be pointed out there is a lot more “theory” behind
the moments of a distribution than is covered here. The depth of discus-
sion about the moments of a distribution presented here will be more than
adequate for our purposes throughout this text.
Thus far, we have covered data distributions in a general sense. Now
we will cover the specific distribution called the Normal Distribution.The Normal Distribution
Frequently, the Normal Distribution is referred to as the Gaussian distri-
bution, or de Moivre’s distribution, after those who are believed to have
discovered it—Karl Friedrich Gauss (1777–1855) and, about a century ear-
lier and far more obscurely, Abraham de Moivre (1667–1754).
The Normal Distribution is considered to be the most useful distribu-
tion in modeling. This is due to the fact that the Normal Distribution ac-
curately models many phenomena. Generally speaking, we can measure
heights, weights, intelligence levels, and so on from a population, and these
will very closely resemble the Normal Distribution.
Let’s consider what is known as Galton’s board (Figure 2.5). This is a
vertically mounted board in the shape of an isosceles triangle. The board is
studded with pegs, one on the top row, two on the second, and so on. Each
row down has one more peg than the previous row. The pegs are arranged
in a triangular fashion such that when a ball is dropped in, it has a 50/50
probability of going right or left with each peg it encounters. At the base of
the board is a series of troughs to record the exit gate of each ball.
The balls falling through Galton’s board and arriving in the troughs
will begin to form a Normal Distribution. The “deeper” the board is (i.e.,
the more rows it has) and the more balls are dropped through, the more
closely the final result will resemble the Normal Distribution.
The Normal is useful in its own right, but also because it tends to be
the limiting form of many other types of distributions. For example, if X is
distributed binomially, then as N tends toward infinity, X tends to be Nor-
mally distributed. Further, the Normal Distribution is also the limiting form
of a number of other useful probability distributions such as the Poisson,
the Student’s, or the T distribution. In other words, as the data (N) used in
these other distributions increases, these distributions increasingly resem-
ble the Normal Distribution.THE CENTRAL LIMIT THEOREMOne of the most important applications for statistical purposes involving
the Normal Distribution has to do with the distribution of averages. The