Weibull distribution function is parameterized as $ F(x) = 1 - e^{(-(x/\sigma)^a)}$ and survival function is $S(x)= e^{(-(x/\sigma)^a)}$ where $a$ is shape parameter and $\sigma$ is the scale parameter.

Mean Residual Function is given by:

$mrl(x) = \frac{\int_{x}^{\infty} S(t) dt}{S(x)} = \frac{\int_{x}^{\infty} e^{(-(t/\sigma)^a)}dt}{e^{(-(x/\sigma)^a)}} $.

I can evaluate the above integral numerically. My question is there a closed form or any analytical form for above mean residual life $mrl(x)$ of Weibull distribution?

  • 1
    $\begingroup$ No, not in the usual meaning of closed form at least, since it involves an incomplete gamma. $\endgroup$
    – Glen_b
    Commented Nov 4, 2022 at 2:36
  • $\begingroup$ @Glen_b, thanks. I have edited my question. Any analytical form would be good as well. $\endgroup$
    – forecaster
    Commented Nov 4, 2022 at 3:25

1 Answer 1


In the interest of having an answer to your question, here's a derivation of an expression for the mean residual life in terms of an incomplete gamma function.

Use a substitution $u = (\frac{t}{\sigma})^a$. Then $du = \frac{a}{\sigma} (\frac{t}{\sigma})^{a-1}dt$.

Since $\frac{t}{\sigma} = u^{1/a}$, this gives us $du = \frac{a}{\sigma} (u^{1/a})^{a-1}dt = \frac{a}{\sigma} u^{1-1/a}dt$, so that $dt = \frac{\sigma}{a} u^{1/a-1} du$. Then

\begin{align*}mrl(x) &= e^{{(x/\sigma)}^a} \int_{(x/\sigma)^a}^{\infty} \frac{\sigma}{a} u^{1/a-1} e^{-u} \, du \\ &= \frac{\sigma}{a} e^{{(x/\sigma)}^a} \Gamma\left(\frac{1}{a}, \left(\frac{x}{\sigma}\right)^a \right), \end{align*} where $\Gamma(s,x)$ is the upper incomplete gamma function.

This can be rewritten slightly by using a recurrence property of the upper incomplete gamma function: \begin{align*} mrl(x) &= \sigma e^{{(x/\sigma)}^a} \left(\Gamma\left(1 + \frac{1}{a}, \left(\frac{x}{\sigma}\right)^a \right) - \left(\left(\frac{x}{\sigma}\right)^a\right)^{1/a} e^{-{(x/\sigma)}^a} \right) \\ &= \sigma e^{{(x/\sigma)}^a} \Gamma\left(1 + \frac{1}{a}, \left(\frac{x}{\sigma}\right)^a \right) - x. \end{align*} The reader can decide which is simpler.

Neither of these expressions is closed-form in the usual sense, as noted in a comment above. However, incomplete gamma functions are implemented in some programming languages like Python. Thus if your interest is numerical you could evaluate the MRL using one of these two expressions and calling that special function rather than using a more general numerical integration procedure.


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