Exponential distribution

The exponential distribution is one the main tools used to model uncertainty, and it is related to other distributions, as well as to an important family of stochastic processes that we will investigate later. An exponential random variable can only take nonnegative values, i.e., its support is [0, +∞), and it owes its name to the functional form of its density:

Fig. 7.12 PDF of symmetric and skewed beta distributions.

Here λ > 0 is a given parameter, and the notation X ∼ exp(λ) is often used.¹² Straightforward integration¹³ yields the CDF

and the expected value is

Fig. 7.13 PDF and CDF of an exponential distribution with λ = 2.

It is worth noting that the expected value is quite different from the mode, which is zero. It can be shown that variance for the exponential distribution is 1/λ², implying that the coefficient of variation is c_X = 1. Figure 7.13 shows the PDF and the CDF for an exponential distribution with parameter λ = 2.

Unlike the uniform distribution, there are typically good physical reasons for adopting this distribution to model a random quantity. A common use of exponential distribution is to model time elapsing between two random events, e.g., the interarrival time between two consecutive service requests. Note that λ is, within this interpretation, a rate at which events occur, e.g., average number of service requests per unit time; the mean interarrival time is 1/λ. In fact, we often speak of exponential random variables with rate λ. There are a few important points worth mentioning:

• The exponential distribution is linked to the Poisson distribution, which we covered in Section 6.5.6. Imagine that the successive interarrival times of service requests are independent¹⁴ and exponentially distributed with rate λ, and count the number of such requests arriving during a time interval of length t. Then, the number of requests we count is a discrete random variable following a Poisson distribution with parameter λt. In Section 7.9 we will see that this phenomenon corresponds to a common stochastic process, which is unsurprisingly known as the Poisson process.
• If we sum n independent exponential variables with rate λ, we obtain a new probability distribution that is called Erlang. This distribution is also widely used in applications to model time between events.
• Probably the most important feature of an exponential random variable is its “lack of memory.” We will consider this property in more detail in Section 8.5.2, but we can realize its intuitive meaning and its practical relevance by considering the waiting time for the arrival of a bus at a bus stop. If we know that the time between two consecutive arrivals is uniformly distributed between, say, 2 and 10 minutes, and we have been waiting for 9 minutes, we may have a pretty clear idea about the time we still have to wait. The more we have waited in so far, the less we are supposed to wait in the future. On the contrary, if this time is exponentially distributed, the fact that we waited for a long time does not change the distribution; the distribution when we get to the bus stop and the distribution after waiting 20 minutes are the same. A full understanding of this requires concepts about independence and conditional distributions, which are provide but it is important to see the practical implication of this property. Imagine that we use the exponential distribution to model time between failures of an equipment. Lack of memory implies that even if the machine has been in use for a long time, this does not mean that it is more likely to have a failure in the near future. Note again the big difference with a uniform distribution. If we know that time between failures is uniformly distributed between, say, 50 and 70 hours, and we also know that 69 hours have elapsed since the last failure, we must expect the next failure within one hour. If the time between failures is exponentially distributed and 69 hours have elapsed, we cannot conclude anything, since from a probabilistic point of view the machine is brand new. If we think of purely random failures, due to bad luck, the exponential distribution may be a plausible model, but definitely it is not if wear is the main driving factor of failures.