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

Overview of Exponential

Distribution

The exponential distribution is used to calculate the probability of occurrence of an event that is the result of a continuous decaying or declining process such as the time until a radioactive particle decays. The exponential distribution is also used to model the waiting times between rare events.

The lengths between arrival times in a Poisson process could be described with the exponential Distribution. Examples of arrival times between Poisson events are as follows:

Time between telephone calls that come over a switchboard

Time between accidents

Time between traffic arrivals

Time between defects

The exponential distribution is not appropriate for predicting failure rates of devices or lifetimes of organisms because a disproportionately high number of failures occur in the very young and the very old. In these cases, the distribution curve would not be a smooth exponential curve as described by the exponential distribution. The Weibull distribution is commonly used to model time-to-failure for devices.

The Exponential Distribution predicts time between Poisson events as follows:

Probability of length of time t between Poisson events = f(t) = λe^-λt

λ (Lamda) is the rate parameter. If, for example, there are 3 events per hour on average, then λ = 3 if time is expressed in units of hours. The exponential distribution calculates the probability of the event occurring at time t given that rate parameter λ = 3 and the occurrence of the event is Poisson-distributed. The average time between events is 1/λ which is 1/3 hours between events.

Graphing the Exponential

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 is the time between events in a Poisson process. The X-Axis expander value is part of this calculation because the width of the exponential PDF graph varies significantly with its Rate Parameter. The X axis must be properly sized to enable the exponential 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 exponential distribution’s PDF value requires the time between Poisson event (the X value) and its only parameter, the Rate parameter λ. The X axis often has to be shifted and expanded in order to view the entire PDF curve fully in a single graph.

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The following Excel-generated graph shows the exponential distribution’s PDF (Probability Density Function) for as the X value (time between Poisson events) goes from 0 to 2.9 with Rate parameter λ = 1.5. This Rate parameter indicates that 1.5 Poisson-distributed events occur on average in each time period.

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 an earlier blog article that provides instructions on how to create an interactive normal distribution PDF curve with outer tails.

Effect of Changing the λ Value

The following Excel-generated graph shows the exponential distribution’s PDF (Probability Density Function) for as the X value (time between Poisson events) goes from 0 to 2.9 with λ increased from 1.5 to 10 Poisson-distributed occurrences per time period on average.

The PDF curve is significantly condensed when λ is increased. When the Rate parameter λ is increased, the events are occurring at a faster average rate and the average time between the events is reduced. The PDF curve of the exponential distribution would be condensed in the rate, λ, is increased as shown in the following Excel-generated graph:

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Effect of Expanding the X Axis

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Graphing the Exponential

Distribution’s CDF – Cumulative

Distribution Function

The following Excel-generated graph shows the Exponential distribution’s CDF (Cumulative Distribution Function) for λ =1.5 Poisson-distributed occurrences per time period on average as the X value (the time between events in a Poisson process) goes from 0 to 2.9.

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. This is shown in the Excel-generated graph below. 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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