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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 thescale 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 distribution function with the same values of as the pdf plots above. | ||||||||||||
Percent Point Function | The formula for the percent point function of the Weibull distribution is The following is the plot of the Weibull percent point function with the same values of as the pdf plots above. | ||||||||||||
Hazard Function | The formula for the hazard function of the Weibull distribution is The following is the plot of the Weibull hazard function with the same values of as the pdf plots above. | ||||||||||||
Cumulative Hazard Function | The formula for the cumulative hazard function of the Weibull distribution is The following is the plot of the Weibull cumulative hazard function with the same values of as the pdf plots above. | ||||||||||||
Survival Function | The formula for the survival function of the Weibull distribution is The following is the plot of the Weibull survival function with the same values of as the pdf plots above. | ||||||||||||
Inverse Survival Function | The formula for the inverse survival function of the Weibull distribution is The following is the plot of the Weibull inverse survival function with the same values of as the pdf plots above. | ||||||||||||
Common Statistics | The formulas below are with the location parameter equal to zero and the scale parameter equal to one.
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Parameter Estimation | Maximum likelihood estimation for the Weibull distribution is discussed in theReliability chapter (Chapter 8). It is also discussed in Chapter 21 of Johnson, Kotz, and Balakrishnan. | ||||||||||||
Comments | The Weibull distribution is used extensively in reliability applications to model failure times. | ||||||||||||
Software | Most general purpose statistical software programs support at least some of the probability functions for the Weibull 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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