The Exponential Distribution

The exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process (the process in which events occur continuously and independently at a constant average rate).

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length (in minutes) of long distance business telephone calls and the amount of time (in months) that a car battery lasts. It could also be shown that the value of the coins in your pocket or purse follows (approximately) an exponential distribution.

Values for an exponential random variable occur in such a way that there are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people that spend less money and fewer people that spend large amounts of money.

Properties of the Exponential Distribution

The mean or expected value of an exponentially distributed random variable $X$, with rate parameter $\lambda$, is given by the formula:

$\displaystyle E[X] = \frac{1}{\lambda}$

Example: If you receive phone calls at an average rate of 2 per hour, you can expect to wait approximately thirty minutes for every call.

The variance of $X$ is given by the formula:

$\displaystyle \text{Var}[X] = \frac{1}{\lambda^2}$

In our example, the rate at which you receive phone calls will have a variance of 15 minutes.

Another important property of the exponential distribution is that it is memoryless. This means that if a random variable $T$ is exponentially distributed, its conditional probability obeys the formula:

$P(T>s+t \ | \ T>s) = P(T>t)$ for all $s, t \geq 0$

The conditional probability that we need to wait, for example, more than another 10 seconds before the first arrival, given that the first arrival has not yet happened after 30 seconds, is equal to the initial probability that we need to wait more than 10 seconds for the first arrival. So, if we waited for 30 seconds and the first arrival didn't happen ($T>30$), the probability that we'll need to wait another 10 seconds for the first arrival ($T>(30+10)$) is the same as the initial probability that we need to wait more than 10 seconds for the first arrival ($T>10$). The fact that $P(T>40 \ | \ T>30) = P(T>10)$ does not mean that the events $T>40$ and $T>30$ are independent.

Applications of the Exponential Distribution

The exponential distribution describes the time for a continuous process to change state. In real-world scenarios, the assumption of a constant rate (or probability per unit time) is rarely satisfied. For example, the rate of incoming phone calls differs according to the time of day. But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives. Similar caveats apply to the following examples which yield approximately exponentially distributed variables:

• the time until a radioactive particle decays, or the time between clicks of a geiger counter
• the time until default (on payment to company debt holders) in reduced form credit risk modeling

Exponential variables can also be used to model situations where certain events occur with a constant probability per unit length, such as the distance between mutations on a DNA strand, or between roadkills on a given road.

In queuing theory, the service times of agents in a system (e.g. how long it takes for a bank teller to serve a customer) are often modeled as exponentially distributed variables. The length of a process that can be thought of as a sequence of several independent tasks is better modeled by a variable following the Erlang distribution (which is the distribution of the sum of several independent exponentially distributed variables).

Reliability engineering also makes extensive use of the exponential distribution. Because of the memoryless property of this distribution, it is well-suited to model the constant hazard rate portion of the bathtub curve used in reliability theory. It is also very convenient because it is so easy to add failure rates in a reliability model. The exponential distribution is, however, not appropriate to model the overall lifetime of organisms or technical devices because the "failure rates" here are not constant: more failures occur for very young and for very old systems.

In hydrology, the exponential distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.