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Overview of the
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 exponential distribution calculates the probability of a specific interval, usually of time, to the first event of a Poisson process. Examples of arrival or waiting times in Poisson processes that could be analyzed with the exponential distribution 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 closely related to the gamma distribution. The exponential distribution calculates the probability of a specific waiting time until the 1st Poisson event occurs. The gamma distribution calculates the probability of a specific waiting time until the kth event Poisson occurs.
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.
Exponential Distribution’s PDF –
Probability Density Function
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. This is shown in the Excel-generated graph below. 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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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)
Characteristics of Poisson-Distributed Events
The Poisson distribution is used for situations that involve counting events over identical intervals of time or objects over identical intervals of volume. If each count is independent of the others, the probability of an event occurring in any of the intervals is constant, and the average count is known, the Poisson distribution can be used to calculate the probability of a specific number of events occurring in an interval.
The Poisson distribution has only one parameter: the rate parameter λ. The rate parameter λ (Lamda) equals the average number of occurrences over the intervals. λ also equals the variance in the number of occurrences over the intervals. One check of whether data are Poisson-distributed is whether the mean number of occurrences equals the variance n the number of occurrences over the intervals.
The Poisson distribution is based upon the following four assumptions:
1) The probability of an event occurring remains constant in all intervals.
2) All events are independent of each other and do not overlap.
3) The probability of observing a single event over a small interval is approximately proportional to the size of that interval.
4) The mean number of occurrences per interval (λ) and the variance in the number of occurrence per interval are approximately the same.
Exponential CDF Problem Solved
in Excel
Customer arrivals at a store’s service desk are Poisson-distributed. On average one customer appears at the customer service desk every 10 minutes. Units of time in which the counts are made are minutes. What is the probability that the wait between two customer arrivals will be UP TO 3 minutes?
The problem asks to calculate the probability that the wait time will be UP TO 3 minutes so the exponential’s CDF (Cumulative Distribution Function) will be used to solve this problem.
If a customer arrives at the service desk on average every 10 minutes, the rate of customer arrivals is 1 customer / 10 minutes or 1/10. The rate parameter λ = 1/10 = 0.10.
The X value to be evaluated is X = 3 minutes
The Excel equation to solve the problem is as follows:
F(X=3;λ=0.1) = EXPON.DIST(X,λ,TRUE) = EXPON.DIST(3,0.1,TRUE) = 0.2592
There is a 25.92 percent probability that the wait between two customer arrivals will be UP TO 3 minutes. This agrees with the CDF graph which X = 3 corresponds with Y = 0.2592 as follows in this Excel-generated graph:
(Click On Image To See a Larger Version)
The exponential distribution equals the gamma distribution when gamma distribution parameters are set as follows:
k = 1
θ = 1/λ
In this case the following is true:
EXPON.DIST(X, λ, FALSE) = GAMMA.DIST(X, 1, 1/λ, FALSE)
and
EXPON.DIST(X, λ, TRUE) = GAMMA.DIST(X, 1, 1/λ, TRUE)
The gamma distribution calculates the probability of wait time for the kth Poisson event. Setting k to 1 configures the gamma distribution to calculate the probability of wait time for the first Poisson event if the average rate is λ. The exponential distribution also calculates the probability of wait time to the first Poisson event when the average rate time is λ. The following two Excel-generated graphs confirm this:
(Click On Image To See a Larger Version)
(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
- Overview of the Normal Distribution
- Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013
- Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013
- Solving Normal Distribution Problems in Excel 2010 and Excel 2013
- Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013
- An Important Difference Between the t and Normal Distribution Graphs
- The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean
- Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way
- t-Distribution in Excel
- Binomial Distribution in Excel
- z-Tests in Excel
- t-Tests in Excel
- Overview of t-Tests: Hypothesis Tests that Use the t-Distribution
- 1-Sample t-Tests in Excel
- Overview of the 1-Sample t-Test in Excel 2010 and Excel 2013
- Excel Normality Testing For the 1-Sample t-Test in Excel 2010 and Excel 2013
- 1-Sample t-Test – Effect Size in Excel 2010 and Excel 2013
- 1-Sample t-Test Power With G*Power Utility
- Wilcoxon Signed-Rank Test As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013
- Sign Test As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013
- 2-Independent-Sample Pooled t-Tests in Excel
- Overview of 2-Independent-Sample Pooled t-Test in Excel 2010 and Excel 2013
- Excel Variance Tests: Levene’s, Brown-Forsythe, and F Test For 2-Sample Pooled t-Test in Excel 2010 and Excel 2013
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro Wilk Tests For Two-Sample Pooled t-Test
- Two-Independent-Sample Pooled t-Test - All Excel Calculations
- 2-Sample Pooled t-Test – Effect Size in Excel 2010 and Excel 2013
- 2-Sample Pooled t-Test Power With G*Power Utility
- Mann-Whitney U Test in Excel as 2-Sample Pooled t-Test Nonparametric Alternative in Excel 2010 and Excel 2013
- 2-Sample Pooled t-Test = Single-Factor ANOVA With 2 Sample Groups
- 2-Independent-Sample Unpooled t-Tests in Excel
- 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013
- Variance Tests: Levene’s Test, Brown-Forsythe Test, and F-Test in Excel For 2-Sample Unpooled t-Test
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk For 2-Sample Unpooled t-Test
- 2-Sample Unpooled t-Test Excel Calculations, Formulas, and Tools
- Effect Size for a 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013
- Test Power of a 2-Independent Sample Unpooled t-Test With G-Power Utility
- Paired (2-Sample Dependent) t-Tests in Excel
- Paired t-Test in Excel 2010 and Excel 2013
- Excel Normality Testing of Paired t-Test Data
- Paired t-Test Excel Calculations, Formulas, and Tools
- Paired t-Test – Effect Size in Excel 2010, and Excel 2013
- Paired t-Test – Test Power With G-Power Utility
- Wilcoxon Signed-Rank Test As a Paired t-Test Alternative
- Sign Test in Excel As A Paired t-Test Alternative
- Hypothesis Tests of Proportion in Excel
- Hypothesis Tests of Proportion Overview (Hypothesis Testing On Binomial Data)
- 1-Sample Hypothesis Test of Proportion in Excel 2010 and Excel 2013
- 2-Sample Pooled Hypothesis Test of Proportion in Excel 2010 and Excel 2013
- How To Build a Much More Useful Split-Tester in Excel Than Google's Website Optimizer
- 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
- Overview of z-Based Confidence Intervals of a Population Mean in Excel 2010 and Excel 2013
- t-Based Confidence Intervals of a Population Mean in Excel 2010 and Excel 2013
- Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean
- Confidence Interval of Population Proportion in Excel 2010 and Excel 2013
- Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013
- Simple Linear Regression in Excel
- Overview of Simple Linear Regression in Excel 2010 and Excel 2013
- Simple Linear Regression Example in Excel 2010 and Excel 2013
- Residual Evaluation For Simple Regression in Excel 2010 and Excel 2013
- Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling Test, and Shapiro-Wilk Test For Simple Linear Regression
- Evaluation of Simple Regression Output For Excel 2010 and Excel 2013
- All Calculations Performed By the Simple Regression Data Analysis Tool in Excel 2010 and Excel 2013
- Prediction Interval of Simple Regression in Excel 2010 and Excel 2013
- Multiple Linear Regression in Excel
- Basics of Multiple Regression in Excel 2010 and Excel 2013
- Multiple Linear Regression Example in Excel 2010 and Excel 2013
- Multiple Linear Regression’s Required Residual Assumptions
- Normality Testing of Residuals in Excel 2010 and Excel 2013
- Evaluating the Excel Output of Multiple Regression
- Estimating the Prediction Interval of Multiple Regression in Excel
- Regression - How To Do Conjoint Analysis Using Dummy Variable Regression in Excel
- Logistic Regression in Excel
- Logistic Regression Overview
- Logistic Regression Performed in Excel 2010 and Excel 2013
- R Square For Logistic Regression Overview
- Excel R Square Tests: Nagelkerke, Cox and Snell, and Log-Linear Ratio in Excel 2010 and Excel 2013
- Likelihood Ratio Is Better Than Wald Statistic To Determine if the Variable Coefficients Are Significant For Excel 2010 and Excel 2013
- Excel Classification Table: Logistic Regression’s Percentage Correct of Predicted Results in Excel 2010 and Excel 2013
- Hosmer-Lemeshow Test in Excel – Logistic Regression Goodness-of-Fit Test in Excel 2010 and Excel 2013
- Single-Factor ANOVA in Excel
- Overview of Single-Factor ANOVA
- Single-Factor ANOVA Example in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group
- Kruskal-Wallis Test Alternative For Single Factor ANOVA in Excel 2010 and Excel 2013
- Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison
- Single-Factor ANOVA - All Excel Calculations
- Overview of Post-Hoc Testing For Single-Factor ANOVA
- Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA
- Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA
- Overview of Effect Size For Single-Factor ANOVA
- ANOVA Effect Size Calculation Eta Squared (?2) in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Psi (?) – RMSSE – in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Omega Squared (?2) in Excel 2010 and Excel 2013
- Power of Single-Factor ANOVA Test Using Free Utility G*Power
- Welch’s ANOVA Test in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar
- Brown-Forsythe F-Test in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar
- Two-Factor ANOVA With Replication in Excel
- Two-Factor ANOVA With Replication in Excel 2010 and Excel 2013
- Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013
- Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication – Test Power With G-Power Utility
- Scheirer-Ray-Hare Test Alternative For 2-Factor ANOVA With Replication
- Two-Factor ANOVA Without Replication in Excel
- Normality Testing in Excel
- Creating a Box Plot in 8 Steps in Excel
- Creating a Normal Probability Plot With Adjustable Confidence Interval Bands in 9 Steps in Excel With Formulas and a Bar Chart
- Chi-Square Goodness-of-Fit Test For Normality in 9 Steps in Excel
- Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk Normality Tests in Excel
- Nonparametric Testing in Excel
- Mann-Whitney U Test in 12 Steps in Excel
- Wilcoxon Signed-Rank Test in 8 Steps in Excel
- Sign Test in Excel
- Friedman Test in 3 Steps in Excel
- Scheirer-Ray-Hope Test in Excel
- Welch's ANOVA Test in 8 Steps Test in Excel
- Brown-Forsythe F Test in 4 Steps Test in Excel
- Levene's Test and Brown-Forsythe Variance Tests in Excel
- Chi-Square Independence Test in 7 Steps in Excel
- Chi-Square Goodness-of-Fit Tests in Excel
- Chi-Square Population Variance Test in Excel
- Post Hoc Testing in Excel
- Creating 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 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
- Solving Problems With Other Distributions in Excel
- Solving Uniform Distribution Problems in Excel 2010 and Excel 2013
- Solving Multinomial Distribution Problems in Excel 2010 and Excel 2013
- Solving Exponential Distribution Problems in Excel 2010 and Excel 2013
- Solving Beta Distribution Problems in Excel 2010 and Excel 2013
- Solving Gamma Distribution Problems in Excel 2010 and Excel 2013
- Solving Poisson Distribution Problems in Excel 2010 and Excel 2013
- Optimization With Excel Solver
- Maximizing Lead Generation With Excel Solver
- Minimizing Cutting Stock Waste With Excel Solver
- Optimal Investment Selection With Excel Solver
- Minimizing the Total Cost of Shipping From Multiple Points To Multiple Points With Excel Solver
- Knapsack Loading Problem in Excel Solver – Optimizing the Loading of a Limited Compartment
- Optimizing a Bond Portfolio With Excel Solver
- Travelling Salesman Problem in Excel Solver – Finding the Shortest Path To Reach All Customers
- Chi-Square Population Variance Test in Excel
- Analyzing Data With Pivot Tables
- SEO Functions in Excel
- Time Series Analysis in Excel
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