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Probability Density Function | The general formula for the probability density function of the exponential distribution is![]() where ![]() ![]() ![]() ![]() ![]() ![]() ![]() 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 |
Weibull Distribution Probability Density Function The formula for the probability density function of the general Weibull distribution is where is the shape parameter , is the location parameter and is the scale parameter . The case where = 0 and = 1 is called the standard Weibull distribution . The case where = 0 is called the 2-parameter Weibull distribution. The equation for the standard Weibull distribution reduces to Since the general form of probability functions can be expressed in terms of the standard distribution , all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the Weibull probability density function. Cumulative Distribution Function The formula for the cumulative distribution function of the Weibull distribution is The following is the plot of the Weibull cumulative ...
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