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Probability Density Function | The general formula for the probability density function of the uniform distribution is where A is the location parameter and (B - A) is the scale parameter. The case where A = 0 and B = 1 is called the standard uniform distribution. The equation for the standard uniform distribution is 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 uniform probability density function. | ||||||||||||||
Cumulative Distribution Function | The formula for the cumulative distribution function of the uniform distribution is The following is the plot of the uniform cumulative distribution function. | ||||||||||||||
Percent Point Function | The formula for the percent point function of the uniform distribution is The following is the plot of the uniform percent point function. | ||||||||||||||
Hazard Function | The formula for the hazard function of the uniform distribution is The following is the plot of the uniform hazard function. | ||||||||||||||
Cumulative Hazard Function | The formula for the cumulative hazard function of the uniform distribution is The following is the plot of the uniform cumulative hazard function. | ||||||||||||||
Survival Function | The uniform survival function can be computed from the uniform cumulative distribution function.The following is the plot of the uniform survival function. | ||||||||||||||
Inverse Survival Function | The uniform inverse survival function can be computed from the uniform percent point function.The following is the plot of the uniform inverse survival function. | ||||||||||||||
Common Statistics |
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Parameter Estimation | The method of moments estimators for A and B are
B = a + h | ||||||||||||||
Comments | The uniform distribution defines equal probability over a given range for a continuous distribution. For this reason, it is important as a reference distribution.One of the most important applications of the uniform distribution is in the generation of random numbers. That is, almost all random number generators generate random numbers on the (0,1) interval. For other distributions, some transformation is applied to the uniform random numbers. | ||||||||||||||
Software | Most general purpose statistical software programs support at least some of the probability functions for the uniform 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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