Example 4.50

There are about 8 million individuals in New York City. How many individuals might we expect to be hospitalized for acute myocardial infarction (AMI), i.e. a heart attack, each day? According to historical records, the average number is about 4.4 individuals. However, we would also like to know the approximate distribution of counts. What would a histogram of the number of AMI occurrences each day look like if we recorded the daily counts over an entire year?

A histogram of the number of occurrences of AMI on 365 days for NYC is shown in Figure 4.12.¹ The sample mean (4.38) is similar to the historical average of 4.4. The sample standard deviation is about 2, and the histogram indicates that about 70% of the data fall between 2.4 and 6.4. The distribution’s shape is unimodal and skewed to the right.

A histogram is shown for "AMI Events (by Day)". There are 11 non-zero values shown: a frequency of about 15 at a value of 1, a frequency of 50 at 2, 70 at 3, 85 at 4, 55 at 5, 45 at 6, 25 at 7, 20 at 8, 5 at 9, 5 at 10, and a frequency of about 2 at 11. — **Figure 4.12.** A histogram of the number of occurrences of AMI on 365 separate days in NYC.

The Poisson distribution is often useful for estimating the number of events in a large population over a unit of time. For instance, consider each of the following events:

having a heart attack,
getting married, and
getting struck by lightning.

The Poisson distribution helps us describe the number of such events that will occur in a day for a fixed population if the individuals within the population are independent. The Poisson distribution could also be used over another unit of time, such as an hour or a week.

The histogram in Figure 4.12 approximates a Poisson distribution with rate equal to 4.4. The rate for a Poisson distribution is the average number of occurrences in a mostly-fixed population per unit of time. In Example 4.50, the time unit is a day, the population is all New York City residents, and the historical rate is 4.4. The parameter in the Poisson distribution is the rate – or how many events we expect to observe – and it is typically denoted by $\lambda$ (the Greek letter lambda) or $\mu$ . Using the rate, we can describe the probability of observing exactly $k$ events in a single unit of time.

Poisson distribution

Suppose we are watching for events and the number of observed events follows a Poisson distribution with rate $\lambda$ . Then $\begin{aligned} P(\text{observe }k\text{ events}) = \frac{\lambda^{k} e^{-\lambda}}{k!} \end{aligned}$ where $k$ may take a value 0, 1, 2, and so on, and $k!$ represents $k$ -factorial, as described on Section 4.3. The letter $e\approx2.718$ is the base of the natural logarithm. The mean and standard deviation of this distribution are $\lambda$ and $\sqrt{\lambda}$ , respectively.

We will leave a rigorous set of conditions for the Poisson distribution to a later course. However, we offer a few simple guidelines that can be used for an initial evaluation of whether the Poisson model would be appropriate.

A random variable may follow a Poisson distribution if we are looking for the number of events, the population that generates such events is large, and the events occur independently of each other.

Even when events are not really independent – for instance, Saturdays and Sundays are especially popular for weddings – a Poisson model may sometimes still be reasonable if we allow it to have a different rate for different times. In the wedding example, the rate would be modeled as higher on weekends than on weekdays. The idea of modeling rates for a Poisson distribution against a second variable such as the day of week forms the foundation of some more advanced methods that fall in the realm of generalized linear models. In Chapters 8 and 9, we will discuss a foundation of linear models.

Exercises

Exercise 4.31

Customers at a coffee shop. A coffee shop serves an average of 75 customers per hour during the morning rush.

Which distribution have we studied that is most appropriate for calculating the probability of a given number of customers arriving within one hour during this time of day?
What are the mean and the standard deviation of the number of customers this coffee shop serves in one hour during this time of day?
Would it be considered unusually low if only 60 customers showed up to this coffee shop in one hour during this time of day?
Calculate the probability that this coffee shop serves 70 customers in one hour during this time of day.

Exercise 4.32

Stenographer’s typos. A very skilled court stenographer makes one typographical error (typo) per hour on average.

What probability distribution is most appropriate for calculating the probability of a given number of typos this stenographer makes in an hour?
What are the mean and the standard deviation of the number of typos this stenographer makes?
Would it be considered unusual if this stenographer made 4 typos in a given hour?
Calculate the probability that this stenographer makes at most 2 typos in a given hour.

Exercise 4.33

How many cars show up? For Monday through Thursday when there isn’t a holiday, the average number of vehicles that visit a particular retailer between 2pm and 3pm each afternoon is 6.5, and the number of cars that show up on any given day follows a Poisson distribution.

What is the probability that exactly 5 cars will show up next Monday?
What is the probability that 0, 1, or 2 cars will show up next Monday between 2pm and 3pm?
There is an average of 11.7 people who visit during those same hours from vehicles. Is it likely that the number of people visiting by car during this hour is also Poisson? Explain.

Exercise 4.34

Lost baggage. Occasionally an airline will lose a bag. Suppose a small airline has found it can reasonably model the number of bags lost each weekday using a Poisson model with a mean of 2.2 bags.

What is the probability that the airline will lose no bags next Monday?
What is the probability that the airline will lose 0, 1, or 2 bags on next Monday?
Suppose the airline expands over the course of the next 3 years, doubling the number of flights it makes, and the CEO asks you if it’s reasonable for them to continue using the Poisson model with a mean of 2.2. What is an appropriate recommendation? Explain.

Notes

These data are simulated. In practice, we should check for an association between successive days. ↩

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