Exponential distribution

The exponential distribution is a class of continuous probability distributions.

It's 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