# 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)

Dan Yarsley says

Hello Sir,

I remember reading an article, which I think you wrote, where it gave a background into how the E term was derived.

Can you please provide me with the link to the article/page please? I cant seem to find it.

Kind fraternal regards.

Fred Schenkelberg says

Hi Dan,

I do not think I’ve written about the derivation of e – e, or Euler’s number, is a mathematical constant, similar to pi. It appears in a range of different applications. For me, e is central to the exponential family of distributions, including the exponential and Weibull distributions.

The exponential decay as seen in a PDF of the exponential distribution arises due to the constant hazard rate. If we start with 100 items and all have the exact same chance of failure per hour, by the time we get to having only 50 items left running, it will take twice as long for the next item to fail (on average) as it took for the first of the 100 to fail. This is because in the first hour we have 100 chances to fail, while in the hour after we have only 50 items left, we only have half as many chances to fail.

hope that helps

cheers,

Fred