The Poisson distribution is a probability distribution for discrete variables over an open-ended unit of measure such as time or space. In other words, it shows the probability of a count of occurrences without a specified number of trials.

There are three main conditions necessary for a Poisson distribution. They are as follows:

1. The number of successes are independent of each other.

2. There is equal probability of success across all units of measurement of equal size.

3. The events occur gradually in succession.

\[P(X=k) = \frac{e^{-\mu} \mu^k}{k!}\] Where (\(\mu\)) = mean number of successes and k = number of times an event occurs in a unit of measurement.

Standard Deviation

\[\sigma = \sqrt{\mu}\]

There are multiple ways to calculate a Poisson distribution probability in R depending on what you’re looking for. Today we will look at functions dpois to find the probability of one singular outcome and ppois to find the cumulative probability of multiple outcomes. Both methods require known values for \(\mu\) and k.

A restaurant receives an average of 4 online orders every hour. What is the probability of it receiving 6 in the next hour?

x = dpois(6,4)
print(paste0("The probability of the restaurant receiving 4 online orders in the next hour is ", round(x,2)))

## [1] "The probability of the restaurant receiving 4 online orders in the next hour is 0.1"

What is the probability of the restaurant receiving 3 or less online orders?

x = ppois(3,4)
print(paste0("The probability of the restaurant receiving at least 3 online orders in the next hour is ", round(x,2)))

## [1] "The probability of the restaurant receiving at least 3 online orders in the next hour is 0.43"

Let’s plot some examples of these calculations to visualize the distribution. A store sells and average of 8 pairs of glasses per day. What is the probability of the store selling 0 through 20 pairs in a day?

A teenager receives an average of 50 texts a day. What is the probability of the teenager receiving less than or equal to 60 texts today?

A manufacturing company has a probability of making 10 errors in every batch. What is the probability of error in any of the 80 products manufactured.

The Poisson Distribution is useful for visualizing probabilities when the number of trials is unspecified and R has many handy tools for the job! Thank you.