Let us better understand the Exponential Distribution formula by working through an example.

Example 37.1

Problem Setup: On a busy Friday evening, the time between customer arrivals at a supermarket check-out counter follows an Exponential distribution. On average, a customer arrives every 30 seconds.

Question: What is the probability a customer will arrive in the next minute at the check-out?

Written solution: Let us use the ‘at most’ formula and follow the steps below:

• Formula: $P(\text{at most or less than}) = P(X \le x) = 1 - e^{-\lambda x}$

• Time units: We will use minutes as the time units.
• Lambda: We convert $\lambda$ to the number of events per minute: $\lambda = \frac{1}{30 \text{ seconds}} \times \frac{60 \text{ seconds}}{1 \text{ minute}}=2 \text{ } \frac{\text{customers}}{\text{per minute}}$
• X-values: We want the probability of the next customer arriving within the next minute. This gives: $P(x \le 1)$.
• e: The other letter, $e$, is a constant. It is the base of the natural log and equal 2.71828.
• Answer: Plugging all of this into the formula $P(X \le x) = 1 - e^{-\lambda x}$ gives:
  $\begin{align} P(X \le 1) &= 1 - e^{-2 \times 1} \\ &= 1 - e^{-2} \\ &= 1 - 0.1353 \\ &= 0.8647 \end{align}$

Conclusion: There is an 86.47% chance that a customer will arrive in the next minute at the check-out.

Next, we will use the formulas to solve for the probability of at least a certain amount of time.

Example 37.2.1

Problem Setup: The times it takes call center specialist to resolve incoming calls to their call center:

• Follow an exponential distribution.
• Have an average of 2.5 minutes.

Question: Specialists will get a warning at 5 minutes to try to wrap up the call (if they can). What is the probability that the specialist receives that 5 minute warning on a call?

You Try: Fill in the values to answer the question in the exercise below:

Answer? Use the values from the above exercise to answer the question. Input your answer in the exercise below:

Let us now explore the memoryless property of the exponential distribution in the next example.

Example 37.2.2

Problem Setup: We will revisit our previous example…

• The times it takes call center specialist to resolve incoming calls follow an exponential distribution.
• On average, it takes the specialist 2.5 minutes to resolve a call.
• After 5 minutes of being on a call, a warning pops up on the specialist’s computer urging them to wrap up the phone call.

Question: What is the probability, after receiving the warning, that it takes at least another 5 minutes to wrap up the call?

Solution: Click here to download the written solutions shown in the video below:

Conclusion: There is 13.53% chance that the call will take at least another 5 minutes to resolve.

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