chapter  25
10 Pages

## - Weibull Distribution

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).