What is exponential distribution

[Greg Bernhardt]

Definition/Summary

The exponential distribution is a probability distribution that describes a machine that it equally likely to fail at any given time.

Equations

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

Extended explanation

A machine is equally likely to fail at any given time. For any $t$, the probability of failure in the interval $(t, t + dt)$ is $\lambda dt$. So the probability that it doesn't fail in that interval must be $1 - \lambda dt$

Let us calculate the probability that it doesn't fail in the interval $(0,t)$. Divide $t$ into $n$ equal parts. Each part then has size $\frac{t}{n}$. The probabilities that it doesn't fail in the intervals $(0,\frac{t}{n})$, $(0,2\frac{t}{n})$, ... are

$$1 - \lambda \frac{t}{n}$$
$$(1 - \lambda \frac{t}{n})^2 ,$$

... respectively. Therefore we find that the probability that it doesn't fail in the interval $(0,t)$ is approximately

$$(1 - \lambda \frac{t}{n})^n .$$

The exact answer is the limit of the above expression as $n \rightarrow \infty$, i.e.

$$e^{-\lambda t}.$$

We can now use this to find the probability density function $f(t)$ that it fails for the first time in the interval $(t,t+dt)$. Clearly, this is the probability that it doesn't fail in the interval $(0,t)$ times the probability that it fails in the interval $(t,t+dt)$.

$$f(t) = e^{-\lambda t} \lambda.$$

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[fresh_42]

The exponential distribution is often used as a model of the duration of random time intervals, such as

• Time between two calls
• Waiting time in a line
• Longevity of atoms in radioactive decay
• Lifetime of components, machinery and equipment when aging phenomena do not need to be considered
• as a rough model for small and medium damages in households
• motor vehicle liability

$\lambda$ represents the number of expected events per unit interval.