Begin2.DVI

The cumulative distribution function F(x)associated with the normal probability

density function f(x) =

1

σ

√

2 π

e−

(^12) (x−μ) (^2) /σ 2

is given by

F(x) =

∫x

−∞

f(x)dx =P(X≤x) (11 .52)

and represents the area under the probability curve form −∞ to x. Note that this

integral cannot be evaluated in a closed form and one must use numerical methods

to calculate the value of the integral for a given value of x. The area calculated

represents the probability P(X≤x). The total area under the normal probability

density function is unity and so the area

1 −F(x) =

∫∞

x

f(x)dx =P(X > x ) (11 .53)

represents the probability P(X > x ). These areas are illustrated in the figure 11-7.

Figure 11-7. Percentage of total area under normal probability curve.

Figure 11-8. Area under normal probability curve representing probabilities.