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 | | | |---|---| | | | | 1\.3.6.6.17. | Beta Distribution | | *[Probability Density Function]()* | The general formula for the [probability density function](https://www.itl.nist.gov/div898/handbook/eda/section3/eda362.htm#PDF) of the beta distribution is f ( x ) \= ( x − a ) p − 1 ( b − x ) q − 1 B ( p , q ) ( b − a ) p \+ q − 1 a ≤ x ≤ b ; p , q \> 0where *p* and *q* are the [shape parameters](https://www.itl.nist.gov/div898/handbook/eda/section3/eda363.htm), *a* and *b* are the lower and upper bounds, respectively, of the distribution, and *B*(*p*,*q*) is the beta function. The beta function has the formulaB ( α , β ) \= ∫ 0 1 t α − 1 ( 1 − t ) β − 1 d tThe case where *a* = 0 and *b* = 1 is called the **standard beta distribution**. The equation for the standard beta distribution isf ( x ) \= x p − 1 ( 1 − x ) q − 1 B ( p , q ) 0 ≤ x ≤ 1 ; p , q \> 0Typically we define the general form of a distribution in terms of location and scale parameters. The beta is different in that we define the general distribution in terms of the lower and upper bounds. However, the location and scale parameters can be defined in terms of the lower and upper limits as follows: Since the general form of probability functions can be [expressed in terms of the standard distribution](https://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm#FORMULAS), all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the beta probability density function for four different values of the shape parameters. | | *[Cumulative Distribution Function]()* | The formula for the [cumulative distribution function](https://www.itl.nist.gov/div898/handbook/eda/section3/eda362.htm#PDF) of the beta distribution is also called the incomplete beta function ratio (commonly denoted by ***Ix***) and is defined as where ***B*** is the beta function defined above. The following is the plot of the beta cumulative distribution function with the same values of the shape parameters as the pdf plots above. | | *[Percent Point Function]()* | The formula for the [percent point function](https://www.itl.nist.gov/div898/handbook/eda/section3/eda362.htm#PPF) of the beta distribution does not exist in a simple closed form. It is computed numerically. The following is the plot of the beta percent point function with the same values of the shape parameters as the pdf plots above. | | *[Other Probability Functions]()* | Since the beta distribution is not typically used for reliability applications, we omit the formulas and plots for the hazard, cumulative hazard, survival, and inverse survival probability functions. | | *Common Statistics* | | | | | | Mean | p p \+ q | | Mode | p − 1 p \+ q − 2 p , q \> 1 | | Range | 0 to 1 | | Standard Deviation | p q ( p \+ q ) 2 ( p \+ q \+ 1 ) | | Coefficient of Variation | q p ( p \+ q \+ 1 ) | | Skewness | 2 ( q − p ) p \+ q \+ 1 ( p \+ q \+ 2 ) p q | | *Parameter Estimation* | First consider the case where *a* and *b* are assumed to be known. For this case, the method of moments estimates are where x ¯ is the sample mean and ***s***2 is the sample variance. If *a* and *b* are not 0 and 1, respectively, then replace x ¯ with x ¯ − a b − a and ***s***2 with s 2 ( b − a ) 2 in the above equations. For the case when *a* and *b* are known, the maximum likelihood estimates can be obtained by solving the following set of equations Maximum likelihood estimation for the case when *a* and *b* are not known can sometimes be problematic. Chapter 14 of [Bury](https://www.itl.nist.gov/div898/handbook/eda/section4/eda43.htm#Bury) discusses both moment and maximum likelihood estimation for this case. | | *Software* | Most general purpose statistical software programs support at least some of the probability functions for the beta distribution. |  - [Site Privacy](https://www.nist.gov/privacy-policy) - [Accessibility](https://www.nist.gov/oism/accessibility) - [Privacy Program](https://www.nist.gov/privacy) - [Copyrights](https://www.nist.gov/oism/copyrights) - [Vulnerability Disclosure](https://www.commerce.gov/vulnerability-disclosure-policy) - [No Fear Act Policy](https://www.nist.gov/no-fear-act-policy) - [FOIA](https://www.nist.gov/foia) - [Environmental Policy](https://www.nist.gov/environmental-policy-statement) - [Scientific Integrity](https://www.nist.gov/summary-report-scientific-integrity) - [Information Quality Standards](https://www.nist.gov/nist-information-quality-standards) - [Commerce.gov](https://www.commerce.gov/) - [Science.gov](https://www.science.gov/) - [USA.gov](https://www.usa.gov/) - [Vote.gov](https://vote.gov/) [](https://www.nist.gov/ "National Institute of Standards and Technology")