2.2.1.3 Poisson Distribution

Poisson Distribution is a distribution that describes the probability of a given number of events occurring within a fixed time interval or region of opportunity (like Distance, Volume, Area, etc.). The distribution has some key characteristics:

• The number of occurrences in each interval range from zero to infinity
• Each event is independent of the other events i.e., the occurrence of one event should not affect the occurrence of a subsequent event
• The average rate (events per interval) is constant.

Say, the number of visitors to a website or the number of calls received by a call center per hour.

Poisson distribution can be calculated through the following formula:

        

Where:

•  is the actual number of successes that occur within a specific interval
•  = (mean), is the number of successes that occur within a specific interval
•  is the Euler's constant, equal to approximately 2.71828

Example 2.3. Imagine a call centre receives 8 calls in an hour, find the probability of the following scenarios:

1. Exactly 10 calls in an hour
2. At least 10 calls in an hour

Here, the values of λ and  are the same for all the scenarios:

•  = 8
•  = 2.71828

Scenario 1: Exactly 10 calls in an hour

Here,  = 10

Substituting the values of , , and  in the Poisson distribution formula yields:

        

Poisson Distribution for this problem can be calculated in excel through the following formula: =POISSON.DIST(, , FALSE) =POISSON.DIST(10, 8, FALSE) = 0.0992

On plotting the same on a chart, we get:

        

The chart can be inferred as follows:

• The x-axis represents the actual number of successes that occur within a specific interval ()
• The y-axis represents the probability of success () for , depicted by bars.
• The Pink Bar represents the probability of exactly 10 successes (i.e., 0.0992, the probability we are looking for)
• All the Red Bars combined with Pink Bar represent the lower cumulative probability, i.e., (  )
• All the Blue Bars combined with Pink Bar represent the upper cumulative probability, i.e., (  )

So, the chart shows that the probability of getting exactly 10 calls in an hour is 0.0992 or 9.92%, which matches with our calculated result.

Scenario 2: At least 10 calls in an hour

Here, we want to calculate for (  ), so essentially we want to know the cumulative probability of getting 10,11, 12, all the way up to infinity calls within one hour (As  increases, the probability will keep on decreasing, becoming almost negligible)

To find the probability distribution for (  ), we can perform the following steps:

• Firstly, instead of calculating for (  ) and going all the way up to infinity, we can go the other way round and find the probabilities of (=), (=),......, (=), (=)
• Add all these probabilities
• Then subtract this number from 1, since the sum of all the probabilities is 1.

So, let's calculate (  ) first:

        

Now, we can simply subtract (  ) from 1:

        

Poisson distribution for this problem can be calculated in excel through the following formula: =1 - POISSON.DIST(, , TRUE) =1 - POISSON.DIST(9, 8, TRUE) = 0.2834

On plotting the same on a chart, we get:

        

The chart can be inferred as follows:

• The x-axis represents the actual number of successes that occur within a specific interval ()
• The y-axis represents the probability of success () for , depicted by bars.
• The pink bar represents the probability of 9 successes
• All the red bars combined with the pink bar represent the lower cumulative probability, i.e., (  )
• All the blue bars combined with the pink bar represent the upper cumulative probability, i.e., (  )

The lower cumulative probability for (  ) is 0.7166, subtracting this probability from 1 gives us the probability as 0.2834. So, the chart shows that the probability of getting at least 10 calls in an hour is 0.2834 or 28.34%, which matches with our calculated result.