## Wednesday, May 28, 2014

### Interactive Graph of Exponential Distribution in Excel 2010 and Excel 2013 (Click On Image To See a Larger Version)

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

Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013

Interactive Graph of the Normal Distribution in Excel 2010 and Excel 2013

Interactive Graph of the Chi-Square Distribution in Excel 2010 and Excel 2013

Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013

Interactive Graph of the t-Distribution’s PDF in Excel 2010 and Excel 2013

Interactive Graph of the t-Distribution’s CDF in Excel 2010 and Excel 2013

Interactive Graph of the Binomial Distribution in Excel 2010 and Excel 2013

Interactive Graph of the Exponential Distribution in Excel 2010 and Excel 2013

Interactive Graph of the Beta Distribution in Excel 2010 and Excel 2013

Interactive Graph of the Gamma Distribution in Excel 2010 and Excel 2013

Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013

# 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. (Click On Image To See a Larger Version)

### 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. (Click On Image To See a Larger Version)

### 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. (Click On Image To See a Larger Version)

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. (Click On Image To See a Larger Version)

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: (Click On Image To See a Larger Version)

### Effect of Expanding the X Axis (Click On Image To See a Larger Version)

## 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)

Excel Master Series Blog Directory

Statistical Topics and Articles In Each Topic

• Histograms in Excel
• Bar Chart in Excel
• Combinations & Permutations in Excel
• Normal Distribution in Excel
• t-Distribution in Excel
• Binomial Distribution in Excel
• z-Tests in Excel
• t-Tests in Excel
• Hypothesis Tests of Proportion in Excel
• Chi-Square Independence Tests in Excel
• Chi-Square Goodness-Of-Fit Tests in Excel
• F Tests in Excel
• Correlation in Excel
• Pearson Correlation in Excel
• Spearman Correlation in Excel
• Confidence Intervals in Excel
• Simple Linear Regression in Excel
• Multiple Linear Regression in Excel
• Logistic Regression in Excel
• Single-Factor ANOVA in Excel
• Two-Factor ANOVA With Replication in Excel
• Two-Factor ANOVA Without Replication in Excel
• Randomized Block Design ANOVA in Excel
• Repeated-Measures ANOVA in Excel
• ANCOVA in Excel
• Normality Testing in Excel
• Nonparametric Testing in Excel
• Post Hoc Testing in Excel
• Creating Interactive Graphs of Statistical Distributions in Excel
• Solving Problems With Other Distributions in Excel
• Optimization With Excel Solver
• Chi-Square Population Variance Test in Excel
• Analyzing Data With Pivot Tables
• SEO Functions in Excel
• Time Series Analysis in Excel
• VLOOKUP

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