In this section, we will study a two-parameter family of distributions that has special importance in reliability.
The function given below is a probability density function for any \(k \gt 0\):
\[ f(t) = k \, t^{k-1} \, \exp(-t^k), \quad 0 \lt t \lt \infty \]The distribution with the density in Exercise 1 is known as the Weibull distribution distribution with shape parameter \(k\), named in honor of Wallodi Weibull. Note that when \(k = 1\), the Weibull distribution reduces to the exponential distribution with parameter 1.
In the special distribution simulator, select the Weibull distribution. Vary the shape parameter and note the shape and location of the density function. For selected values of the shape parameter, run the simulation 1000 times and note the apparent convergence of the empirical density function to the true probability density function.
The following exercise shows why \(k\) is called the shape parameter.
The Weibull probability density function satisfies the following properties:
The distribution function is
\[ F(t) = 1 - \exp(-t^k), \quad 0 \lt t \lt \infty \]The quantile function is
\[ F^{-1}(p) = [-\ln(1 - p)]^{1/k}, \quad 0 \lt p \lt 1 \]In the special distribution calculator, select the Weibull distribution. Vary the shape parameter and note the shape and location of the density function and the distribution function.
With \(k = 2\), find the median and the first and third quartiles. Compute the interquartile range.
The reliability function is
\[ G(t) = \exp(-t^k), 0 \lt t \lt \infty \]The failure rate function is
\[ h(t) = k \, t^{k-1}, \quad 0 \lt t \lt \infty \]The failure rate function \(h\) satisfies the following properties:
Thus, the Weibull distribution can be used to model devices with decreasing failure rate, constant failure rate, or increasing failure rate. This versatility is one reason for the wide use of the Weibull distribution in reliability.
Suppose that \(X\) has the Weibull distribution with shape parameter \(k\). The moments of \(X\), and hence the mean and variance of \(X\) can be expressed in terms of the gamma function.
\(\E(X^n) = \Gamma(1 + \frac{n}{k})\) for \(n \ge 0\).
In the integral for \(\E(X^n)\), use the substitution \(u = t^k\). Simplify and recognize the integral as a gamma integral.
In particular, the mean and variance of \(X\) are
In the special distribution simulator, select the Weibull distribution. Vary the shape parameter and note the size and location of the mean/standard deviation bar. For selected values of the shape parameter, run the simulation 1000 times and note the apparent convergence of the empirical moments to the true moments.
The Weibull distribution is usually generalized by the inclusion of a scale parameter \(b \gt 0\). Thus, if \(Z\) has the basic Weibull distribution with shape parameter \(k\), then \(X = b \, Z\) has the Weibull distribution with shape parameter \(k\) and scale parameter \(b\).
Analogies of the results given above follow easily from basic properties of the scale transformation.
The probability density function is
\[ f(t) = \frac{k}{b^k} \, t^{k-1} \, \exp \left[ -\left( \frac{t}{b} \right)^k \right], \quad 0 \lt t \lt \infty \]Note that when \(k = 1\), the Weibull distribution reduces to the exponential distribution with scale parameter \(b\). The special case \(k = 2\), is called the Rayleigh distribution with scale parameter \(b\), named after William Strutt, Lord Rayleigh.
Recall that the inclusion of a scale parameter does not effect the basic shape of the density; thus the results in Exercise 3 and Exercise 10 hold, with the following exception:
If \(k \gt 1\), the mode occurs at \(t = b \, \left(\frac{k - 1}{k}\right)^{1/k}\).
In the special distribution simulator, select the Weibull distribution. Vary the parameters and note the shape and location of the density function. For selected values of the parameters, run the simulation 1000 times and note the apparent convergence of the empirical density function to the true probability density function.
The distribution function is
\[ F(t) = 1 - \exp\left[-\left(\frac{t}{b}\right)^k\right], \quad 0 \lt t \lt \infty \]The quantile function is
\[ F^{-1}(p) = b [-\ln(1 - p)]^{1/k}, \quad 0 \lt p \lt 1 \]The reliability function is
\[ G(t) = \exp\left[-\left(\frac{t}{b}\right)^k\right], \quad 0 \lt t \lt \infty \]The failure rate function is
\[ h(t) = \frac{k \, t^{k-1}}{b^k}, \quad 0 \lt t \lt \infty \]\(\E(X^n) = b^n \, \Gamma(1 + \frac{n}{k})\) for \(n \ge 0\).
In particular, the mean and variance of \(X\) are
In the special distribution simulator, select the Weibull distribution. Vary the parameters and note the size and location of the mean/standard deviation bar. For selected values of the parameters, run the simulation 1000 times and note the apparent convergence of the empirical moments to the true moments.
The lifetime \(T\) of a device (in hours) has the Weibull distribution with shape parameter \(k = 1.2\) and scale parameter \(b = 1000\).
There is a simple one-to-one transformation between Weibull distributed variables and exponentially distributed variables.
Suppose that \(k \gt 0\) and \(b \gt 0\).
The following exercise is a restatement of the fact that \(b\) is a scale parameter.
Suppose that \(X\) has the Weibull distribution with shape parameter \(k\) and scale parameter \(b\). If \(c \gt 0\) then \(c \, X\) has the Weibull distribution with shape parameter \(k\) and scale parameter \(b \, c\).
Suppose that \((X, Y)\) has the standard bivariate normal distribution. The polar coordinate distance \(R = \sqrt{X^2 + Y^2}\) has the Rayleigh distribution with scale parameter \(\sqrt{2}\).