# 15.1 - Exponential Distributions

15.1 - Exponential Distributions

## Example 15-1

Suppose $$X$$, following an (approximate) Poisson process, equals the number of customers arriving at a bank in an interval of length 1. If $$\lambda$$, the mean number of customers arriving in an interval of length 1, is 6, say, then we might observe something like this:

In this particular representation, seven (7) customers arrived in the unit interval. Previously, our focus would have been on the discrete random variable $$X$$, the number of customers arriving. As the picture suggests, however, we could alternatively be interested in the continuous random variable $$W$$, the waiting time until the first customer arrives. Let's push this a bit further to see if we can find $$F(w)$$, the cumulative distribution function of $$W$$:

Now, to find the probability density function $$f(w)$$, all we need to do is differentiate $$F(w)$$. Doing so, we get:

$$f(w)=F'(w)=-e^{-\lambda w}(-\lambda)=\lambda e^{-\lambda w}$$

for $$0<w<\infty$$. Typically, though we "reparameterize" before defining the "official" probability density function. If $$\lambda$$ (the Greek letter "lambda") equals the mean number of events in an interval, and $$\theta$$ (the Greek letter "theta") equals the mean waiting time until the first customer arrives, then:

$$\theta=\dfrac{1}{\lambda}$$ and $$\lambda=\dfrac{1}{\theta}$$

For example, suppose the mean number of customers to arrive at a bank in a 1-hour interval is 10. Then, the average (waiting) time until the first customer is $$\frac{1}{10}$$ of an hour, or 6 minutes.

Let's now formally define the probability density function we have just derived.

Exponential Distribution

The continuous random variable $$X$$ follows an exponential distribution if its probability density function is:

$$f(x)=\dfrac{1}{\theta} e^{-x/\theta}$$

for $$\theta>0$$ and $$x\ge 0$$.

Because there are an infinite number of possible constants $$\theta$$, there are an infinite number of possible exponential distributions. That's why this page is called Exponential Distributions (with an s!) and not Exponential Distribution (with no s!).

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