Exponential distribution
The exponential distribution is any member of a class of continuous probability distributions assigning probability
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 . Then X follows the exponential distribution with parameter .
