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Probability Density Function | The general formula for the probability density function of the exponential distribution is where is the location parameter and is the scale parameter(the scale parameter is often referred to as which equals ). The case where = 0 and = 1 is called the standard exponential distribution. The equation for the standard exponential distribution is The general form of probability functions can be expressed in terms of the standard distribution. Subsequent formulas in this section are given for the 1-parameter (i.e., with scale parameter) form of the function. The following is the plot of the exponential probability density function. | ||||||||||||||||
Cumulative Distribution Function | The formula for the cumulative distribution function of the exponential distribution is The following is the plot of the exponential cumulative distribution function. | ||||||||||||||||
Percent Point Function | The formula for the percent point function of the exponential distribution is The following is the plot of the exponential percent point function. | ||||||||||||||||
Hazard Function | The formula for the hazard function of the exponential distribution is The following is the plot of the exponential hazard function. | ||||||||||||||||
Cumulative Hazard Function | The formula for the cumulative hazard function of the exponential distribution is The following is the plot of the exponential cumulative hazard function. | ||||||||||||||||
Survival Function | The formula for the survival function of the exponential distribution is The following is the plot of the exponential survival function. | ||||||||||||||||
Inverse Survival Function | The formula for the inverse survival function of the exponential distribution is The following is the plot of the exponential inverse survival function. | ||||||||||||||||
Common Statistics |
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Parameter Estimation | For the full sample case, the maximum likelihood estimator of the scale parameter is the sample mean. Maximum likelihood estimation for the exponential distribution is discussed in the chapter on reliability (Chapter 8). It is also discussed in chapter 19 of Johnson, Kotz, and Balakrishnan. | ||||||||||||||||
Comments | The exponential distribution is primarily used in reliabilityapplications. The exponential distribution is used to model data with a constant failure rate (indicated by the hazard plot which is simply equal to a constant). | ||||||||||||||||
Software | Most general purpose statistical software programs support at least some of the probability functions for the exponential distribution |
Learning Objectives Create and interpret frequency polygons Create and interpret cumulative frequency polygons Create and interpret overlaid frequency polygons Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose as histograms, but are especially helpful for comparing sets of data. Frequency polygons are also a good choice for displaying cumulative frequency distributions . To create a frequency polygon, start just as for histograms , by choosing a class interval. Then draw an X-axis representing the values of the scores in your data. Mark the middle of each class interval with a tick mark, and label it with the middle value represented by the class. Draw the Y-axis to indicate the frequency of each class. Place a point in the
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