Weibull Distribution#

Univariate, Continuous, Asymmetric, Non-Negative, Light-tailed

The Weibull distribution is a continuous probability distribution that models the “waiting time” until an event occurs. It has two parameters: the shape parameter \(\alpha\) and the scale parameter \(\beta\).

This distribution is widely used in fields like engineering, survival analysis, and material science. If \(\alpha > 1\), the event becomes more likely as time passes, making it useful for modeling aging or wear-out processes. If \(\alpha < 1\), the event is more likely at the beginning and decreases over time.

Key properties and parameters#

Support

\(x \in [0, \infty)\)

Mean

\(\beta \Gamma(1 + \frac{1}{\alpha})\)

Variance

\(\beta^2 \Gamma(1 + \frac{2}{\alpha} - \mu^2/\beta^2)\)

Parameters:

  • \(\alpha > 0\): Shape parameter

  • \(\beta > 0\): Scale parameter

Probability Density Function (PDF)#

\[ f(x \mid \alpha, \beta) = \frac{\alpha x^{\alpha - 1} \exp(-(\frac{x}{\beta})^{\alpha})}{\beta^\alpha} \]
/home/docs/checkouts/readthedocs.org/user_builds/preliz/envs/stable/lib/python3.11/site-packages/pytensor/link/numba/dispatch/basic.py:211: UserWarning: Numba will use object mode to run XlogY0's perform method. Set `pytensor.config.compiler_verbose = True` to see more details.
  warnings.warn(

/home/docs/checkouts/readthedocs.org/user_builds/preliz/envs/stable/lib/python3.11/site-packages/pytensor/tensor/xlogx.py:47: RuntimeWarning: divide by zero encountered in log
  return x * np.log(y)
Weibull Distribution PDF

Cumulative Distribution Function (CDF)#

\[ F(x \mid \alpha, \beta) = 1 - \exp(-(\frac{x}{\beta})^{\alpha}) \]
Weibull Distribution CDF

See also

Common Alternatives:

  • Exponential - The Exponential distribution is a special case of the Weibull distribution with the shape parameter \(\alpha = 1\).

Related Distributions:

  • Gumbel - If a random variable follows a Weibull distribution, the logarithm of the random variable follows a Gumbel distribution.

References#