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Probability Density Function | The F distribution is the ratio of two chi-square distributions with degrees of freedom and , respectively, where each chi-square has first been divided by its degrees of freedom. The formula for the probability density function of the F distribution is | ||||||||||||
Cumulative Distribution Function | The formula for the Cumulative distribution function of the F distribution is | ||||||||||||
Percent Point Function | The formula for the percent point function of the F distribution does not exist in a simple closed form. It is computed numerically.The following is the plot of the F percent point function with the same values of and as the pdf plots above. | ||||||||||||
Other Probability Functions | Since the F distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit the formulas and plots for the hazard, cumulative hazard, survival, and inverse survival probability functions. | ||||||||||||
Common Statistics | The formulas below are for the case where the location parameter is zero and the scale parameter is one.
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Parameter Estimation | Since the F distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit any discussion of parameter estimation. | ||||||||||||
Comments | The F distribution is used in many cases for the critical regions for hypothesis tests and in determining confidence intervals. Two common examples are the analysis of variance and the F test to determine if the variances of two populations are equal. | ||||||||||||
Software | Most general purpose statistical software programs support at least some of the probability functions for the F distribution |
Double Exponential Distribution Probability Density Function The general formula for the probability density function of the double exponential distribution is where is the location parameter and is the scale parameter . The case where = 0 and = 1 is called the standard double exponential distribution . The equation for the standard double exponential 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 double exponential probability density function. Cumulative Distribution Function The formula for the cumulative distribution function of the double exponential distribution is The following is the plot of the double exponential cumulative distribution function. Percent Point Function The formula for the percent point function of the double exponential distribution
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