ABSTRACT
Let Y be a standard exponential random variable with probability density function (pdf)
f(y) = e−y, y > 0.
Define X = bY 1/c +m, b > 0, c > 0.
The distribution of X is known as the Weibull distribution with shape parameter c, scale parameter b, and the location parameter m. Its probability density is given by
f(x|b, c,m) = c b
(x−m b
)c−1 exp
{ − [x−m
b
]c} , x > m, b > 0, c > 0. (25.1)
The cumulative distribution function (cdf) is given by
F (x|b, c,m) = 1− exp { − [x−m
b
]c} , x > m, b > 0, c > 0. (25.2)
For 0 < p < 1, the inverse distribution function is
F−1(p|b, c,m) = m+ b(− ln(1− p)) 1c . (25.3) Let us denote the three-parameter distribution by Weibull(b, c,m). A two-parameter Weibull distribution is denoted by Weibull(a, b).
