dfba_beta_descriptive {DFBA} | R Documentation |

## Descriptive Statistics for a Beta Distribution

### Description

Given the two shape parameters for a beta distribution, the function provides central tendency statistics, interval limits, and density and cumulative probabilities.

### Usage

```
dfba_beta_descriptive(a, b, prob_interval = 0.95)
```

### Arguments

`a` |
The first shape parameter for the beta distribution. Must be positive and finite. |

`b` |
The second shape parameter for the beta distribution. Must be positive and finite. |

`prob_interval` |
Desired probability within interval limits (default is .95) |

### Details

The density function for a beta variate is

`f(x) = \begin{cases} Kx^{a-1}(1-x)^{b-1} & \quad \textrm{if } 0 \le x \le 1, \\0 & \quad \textrm{otherwise} \end{cases}`

where

`K = \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)}.`

(Johnson, Kotz, & Balakrishnan, 1995). The two shape parameters `a`

and `b`

must
be positive values.

The `dfba_beta_descriptive()`

function provides features
to complement the beta distribution functions available in the **stats**
package. The function provides the mean, median, mode, and variance for a
beta variate in terms of its two shape parameters.

While the mean, variance, and median are straightforward, there are several
conditions that result in an undefined mode. When either (1) `a = b = 1`

,
(2) `a < 1`

, or (3) `b < 1`

, the mode is undefined. For example,
when `a = b = 1`

, the function is the uniform distribution, which does not
have a modal value. The other cases above result in the density function
diverging at either `x = 0`

or `x = 1`

. The function returns a value of
`NA`

for the mode for all the cases where a unique mode does not exist.

For interval estimation, the function finds an equal-tail interval limits in
all cases, and it also provides the highest-density limits when there is a
well-defined mode. When the mode does not exist, the function returns `NA`

for the limits for the highest-density interval (HDI). For interval
estimation, the probability between the lower and upper limit is the
probability specified in the `prob_interval`

input. The
`dfba_beta_descriptive()`

output object includes a dataframe that has
density and cumulative probability information that can be used for plotting.

### Value

A list containing the following components:

`a` |
The first beta shape parameter |

`b` |
The second beta shape parameter |

`prob_interval` |
The probability for interval estimates |

`x_mean` |
The mean of the distribution |

`x_median` |
The median of the distribution |

`x_mode` |
The mode for the distribution |

`x_variance` |
The variance for the distribution |

`eti_lower` |
The equal-tail lower interval limit |

`eti_upper` |
The equal-tail upper interval limit |

`hdi_lower` |
The lower limit for the highest-density interval |

`hdi_upper` |
The upper limit for the highest-density interval |

`outputdf` |
A dataframe of |

### References

Johnson, N. L., Kotz S., and Balakrishnan, N. (1995). *Continuous Univariate*
*Distributions*, Vol. 1, New York: Wiley.

### See Also

`Distributions`

for additional details on
functions for the beta distribution in the **stats** package.

### Examples

```
dfba_beta_descriptive(a = 38,
b = 55)
dfba_beta_descriptive(38,
55,
prob_interval=.99)
```

*DFBA*version 0.1.0 Index]