Time until an event occurs, or the lifetime of a device

The probability mathematics of the lifetime of machines and devices is the domain of reliability theory.

Reliability theory, at least the elements we consider here, concerns itself with the probability distribution of the time a component or machine will operate before failing. In the simplest case, a device is composed of one simple component, and fails when that component fails. This section looks at the distributions to use to model the lifetime of a single component. We also offer another section that demonstrates how to use the distributions to build the model of the lifetime of a device made of many components. The same distributions are also very useful for modelling the time until some specific event occurs.

First, a little mathematics...

The instantaneous failure rate z(x) of a component is defined as:

                                                                                 

where f(x) and F(x) are the probability density function and cumulative distribution function for x in the usual way. In other words, z(x) is the rate of failure f(x) of the component at time x given that it has survived up to time x with probability 1-F(x).

It can be shown that this expression for z(x) results in an equation for f(x) (the probability density function for the lifetime of the component):

                                                                   (1)

Two interesting results can be obtained from this equation:

The Exponential distribution

If the instantaneous failure rate z(x) is constant i.e. z(x) = 1/b, then putting z(x) into Equation (1) gives:

                           

which is the probability density function of the Exponential distribution, i.e. the Exponential distribution describes the survival time of a component given that it has a constant failure rate. b is often called the mean time between failures (MTBF) in reliability theory parlance.

A constant instantaneous failure rate means that the component 'has no memory', i.e. that it will have no greater or lesser probability of failing at any particular moment no matter how long it has already been running for. In other words, the Exponential distribution would not be appropriate to model a component with either a burn-in period in which it has a high probability of failure, or a component that has a natural limited life, so its probability of failure at any moment increases with time.

b, the mean time between failures, is a scaling factor meaning that changing its value will change the spread of the Exponential distribution but not its shape. That should make sense because, for example, we might choose to measure time in terms of days, weeks, or years but whatever units we use should not change the distribution's shape, although it will obviously change the scale. One way to confirm that is to look at the cumulative distribution function of the Exponential distribution:

From this equation, you can see that multiplying the size of b by 365 (say) would have the same effect as reducing the size of x by a factor of 365, (e.g. changing the units of x from years to days), but the functional relationship remains the same.

The Weibull distribution

If z(t) is not assumed to be constant, but rather increases or decreases smoothly with time, we can consider using the equation:

                                                                               (2)

The equation looks unnecessarily complicated: it is in fact just z(x) = a.tb where a (>0) and b (>-1) are constants, but the form used above helps in producing a neater equation in the next step. If a = 1, the equation for z(x) reduces to: z(x) =1/b which is the formula that produces the exponential distribution. If a is less than 1, z(x) decreases with time which typifies the running in period for a component. If a is greater than 1, z(x) increases with time, which typifies the period of the end of a component's useful life.

Putting the above Equation (2) for z(x) into the f(x) Equation (1) results in the following expression:

            

which is the equation for the Weibull(a,b) distribution. Thus the Weibull distribution is typically used to model the lifetime of a component where its instantaneous failure rate is a function of time. Note that it can only model a time until failure where z(x) is either an increasing function of time or a decreasing function of time, but not both, as shown in the figure below.

The cumulative distribution function for the Weibull is:

It is quite similar to the Exponential distribution for F(x), and we can see that b is again just a scaling factor. However, the a exponent has a very different influence than b. To demonstrate this, let's set b to 1 for convenience (since it is just a scaling factor). If a = 1, we have:

                                                                                        (3)

If a =2 we have:

                                                                                      (4)

If we put values of x = 1, 2, 3 into Equation (4), it would be equivalent to putting values of x = 1, 4, 9 into Equation (3). In other words, an a parameter greater than 1 exaggerates  the life of a component: its as if the component has been working for a lot longer than it really has (compared to an Exponentially distributed time to failure). Similarly, an a value between 0 and 1 is 'shrinking' time.

It looks from the plots of Weibull distributions above that by making a small we reduce the lifetime of the component. However, as a reduces from 1 towards 0, the right tail gets extremely long and the mean time to failure actually gets much larger.

The Lognormal distribution

The Lognormal distribution is also frequently used to model lifetimes of components. It doesn't share the same instantaneous failure rate logical derivation as the Exponential and Weibull. From Central Limit Theorem we know that the product of a large number of random variables can be lognormally distributed. Thus, one can think of the Lognormal distribution as representing that the life of a component is a function of a large number of random factors each of which multiply together to determine the component's actual lifetime.