Failure distributions

A failure distribution is a model that describes mathematically the lifetime of a material, a devise or a structure or system. For instance, it may describe the uncertainty in the amount months before a light bulb burns out.

In this case, let \(X\) be a life variable. An important concept is the failure rate or hazard rate at a time \(t\), denoted as \(r_X(t)\) and given by

\[ r_X(t)=\lim_{\Delta\to 0}\frac{⁡P(t < X ≤t+\Delta t|X>t)}{\Delta t} \]

The failure rate is the probability of the component or system failing at a given instant, the probability of “instantaneous” failure. Note that in the denominator of the equation above, we compute for a given time \(t\) the probability of the lifetime of the component (\(X\)) being higher than \(t\) but smaller than \(t+\Delta\). For small \(\Delta>0\), \(r_X (t)\approx P(X≤t+\Delta|X>t)\). That is, the probability of observing a failure right after time \(t\) given that the component has survived until \(t\).

The failure rate can also be computed with the concepts learned in previous weeks. If the life variable \(X\) has density \(f(t)\) and cumulative distribution function \(F(t)\) then

\[ r_X (t)=\frac{f(t)}{1-F(t)} \]

Some common life distributions used in the literature are Exponential, Gamma and Weibull, as we will see in the next subsections.

Exponential failure rate

Given that the lifetime of a component follows an exponential distribution, \(X \sim Exp(\lambda)\), the failure rate of the variable \(X\) is given by

\[ r(t)=\frac{f(t)}{1-F(t)}=\frac{\lambda e^{-λt}}{1-(1-e^{-\lambda t}) }=λ \]

for \(\lambda>0\) and \(t\geq 0\). Therefore, the failure rate of the variable \(X\) is equal to a constant value \(\lambda\) regardless the instant \(t\). Figure below shows different failure rates associated with the corresponding pdfs of \(t\).

Recall that the PDF of the exponential distribution is given by

\[f(t)=\lambda e^{-\lambda t}\]

while the CDF of the exponential distribution is given by

\[ F(t) = 1-e^{-\lambda t} \]

Fig. 8.9 Exponential distribution for \(t\): (a) pdf, and (b) failure rate.

Gamma failure rate

Given that the lifetime of a component follows an gamma distribution, \(X \sim gamma(\lambda, \alpha)\), the failure rate of the variable \(X\) does not have a closed form but it can be computed using \(r_X (t)=\frac{f(t)}{1-F(t)}\). The density of the Gamma distribution is given by

\[ f(t) = \lambda (\lambda t )^{\alpha -1 } \frac{1}{\Gamma(\alpha)}e ^{-\lambda t} \]

for \(\lambda\), \(\alpha > 0\), \(t\geq 0\) and \(\Gamma\) being the gamma function.

Figure below depicts the influence of the parameters of the gamma distribution , \(\lambda\) and \(\alpha\) in their corresponding failure rate. When \(\alpha=1\), the gamma distribution reduces to the exponential and, thus, the failure rate is constant; \(\alpha\) controls the shape of the distribution and, thus, the shape of the failure rate. The gamma distribution has increasing failure rate for \(\alpha >1\). To get an intuition of the influence of the parameters \(\alpha\) and \(\lambda\) in the failure rate, you can play with the interactive element below.

Fig. 8.10 Gamma distribution for \(t\): (a) pdf, and (b) failure rate.

Weibull failure rate

Given that the lifetime of a component follows an Weibull distribution, \(X \sim Weibull(\lambda, \alpha)\), the failure rate of the variable \(X\) is given by

\[ r(t) = \lambda \alpha t^{\alpha-1} \]

for \(\lambda\), \(\alpha > 0\), \(t\geq 0\).

Figure below depicts the influence of the parameters of the Weibull distribution , \(\lambda\) and \(\alpha\) in their corresponding failure rate.

Fig. 8.11 Weibull distribution for \(t\): (a) pdf, and (b) failure rate.

The Weibull distribution has increasing failure rate for \(\alpha >1\). Similarly to the gamma distribution, \(\alpha\) controls the shape of the distribution and, thus, the shape of the failure rate. To get an intuition of the influence of the parameters \(\alpha\) and \(\lambda\) in the failure rate, you can play with the interactive element below.

Notice that some common parameterizations for the Weibull density are

\[ f(t) = \lambda\alpha t^{\alpha-1}e^{-\lambda t^\alpha} \]

\[ f(t) = \frac{b}{a}\left(\frac{t}{a}\right)^{b-1} e^{-\left(\frac{t}{a}\right)^b} \]

for \(a\), \(b > 0\), \(t\geq 0\). \(a\) is usually called the scale parameters and \(b\), the shape parameter.

Play with the parameters!

Fig. 8.12 Interactively visualize the influence of the parameters in the failure rate for the Exponential, Gamma and Weibull distribution.

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Attribution

This chapter is written by Patricia Mares Nasarre, Max Ramgraber and Oswaldo Morales Napoles. Find out more here.