The Weibull Distribution
The probability density function of a Weibull random variable X is
whereis the shape parameter andis the scale parameter. This is shown below.
Ifis a "timetofailure", the distribution gives a distribution for which the failure rate is proportional to a power of time. The failure rate here isThe shape parameter,is that power plus one, and this parameter can be interpreted directly as follows:

k<1 indicates that the failure rate decreases over time. This indicates a high initial failure rate. As these initial failures are overcome, only the 'healthy' long lived ones remain.

indicates that the failure rate is constant, suggesting failure is due to random events.

Indicates the failure rate increases over time, indicating an aging process as with many manufactured goods.
The cumulative distribution function is
The mean and variance of the Weibull distribution are andrespectively.
The moment generating function can best be expressed as the integral
This integral is not elementary.