Ralph Vince - Portfolio Mathematics

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

Probability Distributions 55

FIGURE 2.6 The Exponential Distribution and the Normal

where: U=The mean of the data.

S=The standard deviation of the data.

X=The observed data point.

EXP( )=The exponential function.

This formula will give us the Y axis value, or the height of the curve if you

will, at any given X axis value.

Often, it is easier to refer to a point along the curve with reference to

its X coordinate in terms of how many standard deviations it is away from

the mean. Thus, a data point that was 1 standard deviation away from the

mean would be said to be onestandard unitfrom the mean.

Further, it is often easier to subtract the mean from all of the data

points, which has the effect of shifting the distribution so that it is centered

over zero rather than over the mean. Therefore, a data point that was 1

standard deviation to the right of the mean would now have a value of 1 on

the X axis.

When we make these conversions, subtracting the mean from the

data points, then dividing the difference by the standard deviation of the

data points, we are converting the distribution to what is called thestan-

dardized normal, which is the Normal Distribution with mean=0 and