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10 THE HANDBOOK OF PORTFOLIO MATHEMATICS
Notice here thatif we were to plot the asterisks vertically we would
be developinginto the familiar bell-shaped curve, also called the Normal
or Gaussian Distribution (see Figure 1.1).^1FIGURE 1.1 Normal probability function(^1) Actually, the coin toss does not conform to the Normal Probability Functionina
pure statistical sense, but rather belongs to a class of distributions called the Bi-
nomial Distribution (a.k.a.Bernoullior Coin-Toss Distributions).However, as N
becomes large, the Binomial approaches the Normal Distribution as a limit (pro-
vided the probabilitiesinvolved are not close to 0 or 1).Thisis so because the
Normal Distributionis continuous from left to right, whereas the Binomialis not,
and the Normalis always symmetrical whereas the Binomial needn’tbe.Since we
are treatingafinite number of coin tosses and tryingto make them representative
of the universe of coin tosses, and since the probabilities are always equal to.5,
we will treat the distributions of tosses as though they were Normal.As a further
note, the Normal Distribution can be used as an approximation of the Binomialif
both N times the probability of an event occurringand N times the complement of
the probability occurringare bothgreater than 5.In our coin-toss example, since
the probability of the eventis.5 (for either heads or tails) and the complementis
.5, then so longas we are dealingwith N of 11 or more we can use the Normal
Distribution as an approximation for the Binomial.