• It is important to stress that we must always match the time units.
• In the example from the previous section, the average arrival rate was given per minute.
• The question’s time interval was also one minute: “What is the probability of 5 customers arriving in any given minute.”
• So, no adjustment of lambda ($\lambda$) was needed since the time intervals matched.
• See the next example to see where an adjustment of $\lambda$ is needed.

Example 32.1

Problem Setup: The Canadian Anti-Fraud Centre received an average of 3,578 reports of fraud per month so far in 2024, with losses totaling $123,000,000 in the first quarter of 2024 alone!

Question: If we assume that the number of frauds reported to the Canadian Anti-Fraud Centre follows a Poisson distribution, what is the probability of there being at most 3,578 frauds reported in the month of May?

Solutions: Click here to download the Excel solutions shown in the video below:

Conclusion: We use =POISSON.DIST(3578, 3578, TRUE). This gives a 50.44% chance that the number of reports received by the Anti-Fraud Centre in the month of May do not exceed the monthly average for 2024 to date.

Let us now review how to calculate the probability of more than $x$ events occurring during a certain time frame in the exercise in the next example.

Example 32.2

Problem Setup: The Canadian Anti-Fraud Centre reports an average of 2,636 victims of fraud per month so far in 2024. They reported a total of 41,988 victims in 2023 (or 3,499 victims per month).  Let us assume that the number of victims of fraud in 2024 follows a Poisson distribution with an average of 2,636 victims per month.

Question: What is the probability of there being more than 3,499 victims of fraud in any given month in 2024? Ie: What is the probability that any given month in 2024, the number of victims of fraud will exceed the monthly average for 2023?

You Try: Solve the above problem by placing the values in the correct positions below:

• Finally, we will practice calculating the probability of ‘at least’ $x$ events in a certain time-frame.
• See the next example for a video walk-through of how to calculate $P(X \ge x)$ in Excel.

Example 32.3

Problem Setup: There are, on average, 28,000 bankruptcies filed in Canada per year according to the Office of the Superintendent of Bankruptcy in Canada.

Question: If the number of bankruptcies follows a Poisson distribution, what is the probability of at least 2,400 bankruptcies getting filed in any given month in Canada?

Solution: Use = 1− POISSON.DIST(2399, 28000/12, TRUE). Click here to download the Excel solutions shown below:

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