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probabilistic Design analysis • 89
where g is the shape parameter, m is the location parameter, and a is
the scale parameter. The case where m = 0 and a = 1 is called the standard
lognormal distribution. The case where m = 0 and a = 1 is called the
2-parameter Weibull distribution. The equation for the standard Weibull
distribution reduces to:

fx () = () (g − ) 1 exp − ( ) ≥ 0; g > 0
g
g
xx
x

Weibull PDF (g = 0.5) Weibull PDF (g = 1)
1 0.6
0.8
Probability 0.5 Probability 0.4
0.2
0 0
0 1 2 3 4 5 0 1 2 3 4 5
x x
Weibull PDF (g = 2) Weibull PDF (g = 5)
0.8 1
0.6
Probability 0.4 Probability 0.5
0.2
0 0
0 1 2 3 4 5 0 1 2 3 4 5
x x
Figure 3.8. The Weibull probability density function for four values of g.

Since the general form of probability functions can be expressed in
terms of the standard distribution, all subsequent formulas in this section
are given for the standard form of the function.
Figure 3.8 shows the Weibull pdf for four values of g.

3.2.5.2 Cumulative Distribution Function

The formula for the cdf of the Weibull distribution is:

g
Fx () =− e − x ( ) x ≥ 0 g > 0
;
1
Figure 3.9 shows the Weibull cdf with the same values of as the pdf
plots above.