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QUAN 2010 Notes: Introduction to Business Statistics

Omar Ortega

Section 7.2 The Exponential Distribution

Definition 7.2.1.

The exponential probability distribution is a continuous distribution commonly used in business to measure the time between customer arrivals or the time between failures in a business process
The mathematical expression that describes the shape of the curve for the exponential probability distribution is called the exponential probability density function:
\begin{equation*} f(x)=\lambda^{-\lambda x}\text{, where }\lambda=\text{the mean number of occurrences over an interval} \end{equation*}
Recall the we saw \(\lambda\) in a previous chapter when the discrete Poisson distribution was introduced. It is not a coincidence that we are using \(\lambda\) in both of these distributions. A discrete random variable that follows a Poisson distribution with a mean equal to \(\lambda\) has a counterpart continuous random variable that follows the exponential distribution with a mean equal to \(\mu=\frac{1}{\lambda}\text{.}\)
It is easy to confuse the two -- try to remember that \(\lambda\) is a countable (discrete) rate while \(\mu\) is a measurable (continuous) interval.
A small bit of good news: the standard deviation for an exponential distribution is equal to its mean. That is, \(\sigma=\mu=\frac{1}{\lambda}\text{.}\)
The only parameter in the formula is \(\lambda\text{.}\) Let’s look at how it impacts the shape of the distribution.
Figure 7.2.2. Exponential Distribution powered by Desmos

Subsection 7.2.1 Differences between the Exponential and Normal Distributions

  1. The shape of the exponential distribution is right-skewed while the shape of the normal distribution is bell-shaped and symmetric.
  2. The shape of the exponential distribution is controlled by only one parameter, \(\mu\text{,}\) while the shape of the normal distribution requires two parameters, \(\mu\) and \(\sigma\text{.}\)
  3. The values of an exponential distribution are always positive; they can never be negative. Values of the normal distribution can be both positive and negative.

Subsection 7.2.2 Computing Probabilities for the Exponential Distributions

There is no table for helping us calculate probabilities associated with an exponential distribution. So we will rely on a formula and Excel to help with these calculations. The formula for calculating the probability that a random variable is less than a specified value is:
\begin{equation*} P(x\leq a)=1-e^{-a\lambda} \end{equation*}
where \(\lambda=\text{the mean number of occurrences over an interval}\text{.}\)
The Excel formula for calculating probabilities for the exponential distribution is
\begin{equation*} EXPON.DIST(x,\lambda,\text{cumulative}), \end{equation*}
where cumulative=TRUE (if you want cumulative probability) or FALSE (if you do not want cumulative probability).

Exercise 7.2.3.

(Donnelly 6.21,22)
An exponential probability distribution has a mean equal to 5 minutes per customer. Calculate the following probabilities for the distribution.
(a)
\(P(X\leq 10)\)
Answer.
\(\mu=\frac{5\text{ min}}{1\text{ customer}}\Rightarrow \lambda=\frac{1\text{ customer}}{5\text{ min}}=0.2\)
\(1-e^{-10(.2)}\approx 0.8647\) OR \(EXPON.DIST(10,0.2,1)\)
(b)
\(P(X\gt 12)\)
Answer.
\(1-P(X\lt 12)=1-EXPON.DIST(12,.2,1)\approx 0.0907\)
(c)
\(P(1\leq X\leq 5)\)
Answer.
\(P(X\leq 5)-P(X\leq 1)\) \(=EXPON.DIST(5,.2,1)-EXPON.DIST(1,.2,1)\) \(\approx 0.4509\)

Exercise 7.2.4.

An exponential probability distribution has lambda equal to \(18\) customers per hour. Find the probability that the next customer will arrive within the next minute. (Round your answer to 4 decimal places.)

Exercise 7.2.5.

An exponential probability distribution has lambda equal to \(18\) customers per hour. Find the probability that the next customer will arrive within the next 15 minutes. (Round your answer to 4 decimal places.)

Exercise 7.2.6.

An exponential probability distribution has lambda equal to \(18\) customers per hour. Find the probability that the next customer will arrive within the next 2 to 5 minutes. (Round your answer to 4 decimal places.)

Exercise 7.2.7.

An exponential probability distribution has lambda equal to \(18\) customers per hour. Find the probability that the next customer will arrive within the next 45 seconds. (Round your answer to 4 decimal places.)
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