Weibull Distribution

Modeling time-to-failure, event durations, and reliability.

The "Time-to-Event" Distribution

The Weibull distribution is a highly flexible continuous probability distribution. It's widely used in engineering to model reliability and time-to-failure of components. In finance, it can be applied to model the duration of events, such as the time until a corporate bond defaults or the time a stock price stays above a certain level.

Its flexibility comes from its shape parameter, $k$ . Depending on the value of $k$ , it can mimic the behavior of other distributions like the exponential (when $k=1$ ) or approximate the normal distribution (when $k$ is around 3-4).

The Formula

The probability density function (PDF) is given by:

f(x; k, \lambda) = \frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k}

$x \ge 0$ is the variable (e.g., time).
$k > 0$ is the shape parameter. It determines the shape of the failure rate. If $k < 1$ , the failure rate decreases over time. If $k = 1$ , it's constant (Exponential). If $k > 1$ , the failure rate increases over time (wear-out).
$\lambda > 0$ is the scale parameter, which stretches or contracts the distribution.

Interactive Weibull Distribution

Adjust the shape (k) and scale (λ) parameters to see how the distribution's form changes.

Shape (k): 2.0

Scale (λ): 1.0