Exponential distribution

The exponential distribution is any member of a class of continuous probability distributions assigning probability

$e^{-x/\mu }\,$

to the interval [</nwoiki>''x'', ∞<nowiki>).

It is well suited to model lifetimes of things that don't "wear out", among other things.

The exponential distribution is one of the most important elementary distributions.

A basic introduction to the concept

The main and unique characteristic of the exponential distribution is that the conditional probabilities P(X>x+1 given X>x) stay constant for all values of x.

More generally, we have P(X>x+s given X>x)= P(X>s) for all x and s.

Example

A living person's final total length of life may be represented by a stochastic variable X.

A newborn will have a certain probability of seeing his 10th birthday, a 10 year old will have a certain probability of seeing his 20th birthday, and so on. Regrettably, a 60 year old may count on a slightly smaller probability of seeing his 70th birthday, and an octogenarian's chances of enjoying 10 more years may be smaller still.

So in the real world, X is not exponentially distributed. If it were, all probabilities mentioned above would be identical.

Formal definition

Let X be a real, positive stochastic variable with probability density function $f(x)=\lambda e^{-\lambda x},\lambda \in <0,\infty >$. Then X follows the exponential distribution with parameter $\lambda$.

See also

• Continuous probability distribution
• Probability
• Probability theory

Related topics

• Poisson distribution