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This is one of the following eleven articles on creating user-interactive graphs of statistical distributions in Excel

Poisson Distribution

Overview

The Poisson distribution is a family of discrete probability distributions. This is evidenced by the stepwise shape of the above graph of a Poisson distribution’s PDF (Probability Density Function) curve. The stepwise shape of a discrete distribution indicates that the discrete distribution can only assume discrete values and is not continuous.

The Poisson distribution has one parameter, λ (Lamda). λ equals the average number of events occurring in a given unit of time. The Poisson Distribution is used to calculate the probability of a specific number of events occurring over a unit of time if the average number of events occurring over that unit of time equals λ and the occurrence of the event is distributed according to the Poisson distribution.

Previous measurement must have been taken to determine the following:

1) The events occur in frequency according to the Poisson distribution

2) The average rate, which is the expected number of occurrences of that event over the given time period.

The PDF (Probability Density Function) of the Poisson distribution predicts the degree of spread around a known average rate of occurrence.

Examples of events whose frequency of occurrence over a given period of time are often distributed according to the Poisson distribution are the following:

Number of telephone calls that come over a switchboard

Number of cars arriving at a traffic light

Number of accidents at an intersection

Number of customers arriving at a sales counter

Number of insurance losses/claims filed

Number of goals in sports involving two competing teams

Number of jumps in stock price

Number of times a web server is accessed

Graphing the Poisson

Distribution’s PDF –

Probability Density Function

Step 1 – Create a Count

The Count becomes the basis for the X and Y values of each data point on the graph. This count will contain 100 points that count from 0 to 100 in increments 1. There are many ways to create a count. This count uses the method ROW() – ROW($B) to increment each cell value in the column by 1.

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Step 2 – Create the X Values

The X values for each data point are created. The X-Axis expander and X-Axis shifter values are part of this calculation because the location and width of the normal distribution’s PDF graph varies significantly when its only parameter λ change. The X axis must be properly sized to enable the Poisson distribution’s PDF curve to be fully visible and fully expanded in the graph.

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Step 3 – Create the Y Values

One Y value is created for each increment of the Count. The Y value of each data point is its PDF value. The Poisson distribution’s PDF value requires its only parameter λ.

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The following Excel-generated graph shows the normal distribution’s PDF (Probability Density Function) for as the X value goes from 2 to 15 with λ = 10.

The PDF value of a statistical distribution (the Y value) at a specific X value equals the probability that the value of a random sample will be equal to that X value if the population of data values from which the sample was taken is distributed according the stated distribution. The CDF value of a statistical distribution (the Y value) at a specific X value equals the probability that the value of a random sample will be up to that X value.

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The process of creating an Excel area chart and connecting the user inputs to the chart is shown in detail in the section of this manual that provides instructions on how to create an interactive normal distribution PDF curve with outer tails.

Effect of Changing λ

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Effect of Changing the X-Axis Shifter

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Effect of Changing the X-Axis Expander

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Graphing the Poisson

Distribution’s CDF –

Cumulative Distribution Function

The following Excel-generated graph shows the Poisson distribution’s CDF (Cumulative Distribution Function) for λ = 10 as the X value goes from 2 to 35.

The CDF value of a statistical distribution (the Y value) at a specific X value equals the probability that the value of a random sample will be up to that X value if the population of data values from which the sample was taken is distributed according the stated distribution. The PDF value of a statistical distribution (the Y value) at a specific X value equals the probability that the value of a random sample will be equal to that X value.

(Click on Image To See a Larger Version)

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