The Exponential Distribution

The exponential distribution is a model for items with a constant failure rate (which very rarely occurs).

If the chance of failure is the same each hour (or cycle, etc.), including the first hour, 100th hour, and 1 millionth hour or use, then the exponential distribution is suitable.

Many practitioners assume the failure rate either is constant or changes so little as to be essential constant in order to make reliability calculations simpler. The assumption is so common, and often inaccurate, that you should be aware of the exponential distribution. Furthermore, many questions in the CRE exam use the exponential distribution.

The exponential distribution is related to the Poisson distribution. For example, if a random variable, x, is exponentially distributed then the reciprocal follows a Poisson distribution. If x = 1 / y then y is a Poisson distribution.

The exponential distribution is a reasonable model for the mean time between failures, such as failure arrivals, whereas the Poisson distribution is used to model the number of occurrences in an interval.

Probability Density Function (PDF)

The exponential distribution PDF is similar to a histogram view of the data and expressed as

$$ \large\displaystyle f\left( x \right)=\frac{1}{\theta }{{e}^{-{}^{x}\!\!\diagup\!\!{}_{\theta }\;}}=\lambda {{e}^{\lambda x}}$$

Where, $- \lambda -$ is the failure rate and $- \theta -$ is the mean

Keep in mind that

$$ \large\displaystyle \lambda =\frac{1}{\theta }$$

Cumulative Density Function (CDF)

The CDF is the integral of the PDF and expressed as

$$ \large\displaystyle F\left( x \right)=1-{{e}^{-\lambda x}}=1-{{e}^{-{}^{x}\!\!\diagup\!\!{}_{\theta }\;}}$$

And represents the probability of failure over x which is commonly a duration. Of course, x ≥ 0.

Reliability Function

The reliability or probably of success over a duration, x, is

$$ \large\displaystyle R\left( x \right)={{e}^{-\lambda x}}={{e}^{-{}^{x}\!\!\diagup\!\!{}_{\theta }\;}}$$

And x ≥ 0 applies here as well.

Hazard Function

The exponential hazard function is determined via the ration of the PDF and Reliability functions

$$ \large\displaystyle h\left( x \right)=\frac{f\left( x \right)}{R\left( x \right)}=\frac{\lambda {{e}^{-\lambda x}}}{{{e}^{-\lambda x}}}=\lambda $$

Which is a constant.

Interesting Properties

The exponential distribution has a few interesting properties, primarily that it is memoryless. In other words, the probability of failure in the next instant or hour does not depend on how old the item is nor how long it has been in service.
Another aspect to consider is it is possible to add failure rates. This makes parts count predictions possible using limited tools for calculations. It used to be very easy to add and much more difficult to multiply or apply exponents, thus the desire to assume a constant failure rate.

Related:

Weibull Distribution (article)

Common formulas (article)

The Poisson Distribution (article)