Exponential Distribution

The exponential distribution is a continuous probability distribution which describes the amount of time it takes to obtain a success in a series of continuously occurring independent trials. It is a continuous analog of the geometric distribution.

The Poisson distribution is a discrete distribution modeling the number of times an event occurs in a time interval, given that the average number of events \(\lambda\) occurring in the interval is known. In particular, if \(X\) is Poisson distributed, with mean \(\lambda\), then

\[P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}.\]

Many real-life processes can be modeled as Poisson processes. For instance, the number of goals scored during a soccer match, or the number of patients visiting a doctor's office during a particular hour, can both be thought of as Poisson-distributed random variables.

Now, suppose \(X\) is Poisson-distributed, with mean \(\lambda\), and assume it describes the arrival of patients to a doctor's office over the time interval \([0,1]\). Then, there is a collection of random variables \(X_t\) indexed by \(t\in [0,1]\), where \(X_t\) denotes the number of arrivals in the interval \([0,t]\). Each \(X_t\) is ostensibly Poisson-distributed, with mean \(t\lambda\). In particular, a feature of the Poisson process is that it is translation-invariant: the number of patient arrivals only depends on the length of the time interval in which they occur, and not on when the interval begins.

For each \(t\in [0,1]\) define a continuous random variable \(Y_t\) representing the amount of time it takes for the next patient to arrive, given that one arrived at time \(t\). Then, the event \((Y_t > x)\) is precisely the same as the event \((X_{t} = X_{t+x})\). Hence,

\[P(Y_t > x) = P(X_t = X_{t+x}) = P(X_0 = X_{x}) = P(X_{x} = 0) = e^{-x\lambda}.\]

Note that all the random variables \(Y_t\) have the same distribution, unlike the \(X_t\)'s. The distribution of \(Y_t\) is called the exponential distribution. In this context, the number \(\lambda\) is called the rate parameter of the exponential distribution.

More generally, the exponential distribution can be thought of as describing a continuous analogue of the geometric distribution. A geometrically distributed discrete random variable describes the number of trials one needs to get a success; similarly, the exponential distribution, as determined above, describes the amount of time one needs to get a success.

The exponential distribution with rate parameter \(\lambda\) is denoted \(\mathcal{E}(\lambda)\). From the above computation, it is known that the cumulative density function of this distribution is

\[f_{\lambda} (x) = \left\{ \begin{array}{ll} 1 - e^{-\lambda x} & : x \ge 0\\ 0 & : x < 0. \end{array} \right.\]

Differentiating this shows that the probability density function is

\[p_{\lambda} (x) = \left\{ \begin{array}{ll} \lambda e^{-\lambda x} & : x \ge 0\\ 0 & : x < 0. \end{array} \right.\]

Any real-life process consisting of infinitely many continuously occurring trials could be modeled using the exponential distribution. Whether or not this model is accurate will depend on if the assumption of a constant rate at which successes occur is valid. In practice, this is often not the case: for example, it may be twice as likely to receive a phone call between 6:00 PM and 7:00 PM than between 10:00 PM and 11:00 PM.

However, if the time interval considered does have an approximately constant rate at which phone calls occur (say, the time between 6:00 PM and 7:00 PM; it is reasonable to assume that the rate at which phone calls occur will be approximately constant throughout this hour), then the time until the next phone call is justifiably modeled as an exponentially distributed random variable.