0.4
0.2
-3 -2 -1 0 1 2 3 4 5 6 7
Figure 3-11. Probability Density Function for the Normal Distribution, N: (2, 2)
For each value of the normal deviate x, there is an associated density of probability
f(x) and the probability of one value lying between x and x+dx will bef(x) dx.
Consequently, the probability ofx lying between a and b will be equal to:
proba < x < b}= f(x)dx (3-46)
ra
And this integral, if extended from -oo to +oo is also equal to:
prob{- o < x < +c}= +c f (x)dx = 1 (3-47)
The probability ofx being smaller than or equal to a given value ofxo is called the
cumulative probability, F(xo), and is equal to:
F(xo) = prob{x < x, }= f (x)dx (3-48)
By the definition, cumulative probability can be written as:
F (-o) 0 and F (+o) =1 (3-49)
The integral ff f(x)dx for a normally distributed deviate is not easily solved.
Below is an approximation and was used herein to generate a normal distribution of log-