# Binomial distribution for fitting a GAMLSS

### Description

The `BI()`

function defines the binomial distribution, a one parameter family distribution, for a `gamlss.family`

object to be used
in GAMLSS fitting using the function `gamlss()`

.
The functions `dBI`

, `pBI`

, `qBI`

and `rBI`

define the density, distribution function, quantile function and random
generation for the binomial, `BI()`

, distribution.

### Usage

1 2 3 4 5 |

### Arguments

`mu.link` |
Defines the |

`x` |
vector of (non-negative integer) quantiles |

`mu` |
vector of positive probabilities |

`bd` |
vector of binomial denominators |

`p` |
vector of probabilities |

`q` |
vector of quantiles |

`n` |
number of random values to return |

`log, log.p` |
logical; if TRUE, probabilities p are given as log(p) |

`lower.tail` |
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x] |

### Details

Definition file for binomial distribution.

*f(y|mu)=(Gamma(n+1)*Gamma(y+1)/Gamma(n-y+1))* mu^y *(1-mu)^(n-y)*

for *y=0,1,2,...,n* and *0<μ< 1*.

### Value

returns a `gamlss.family`

object which can be used to fit a binomial distribution in the `gamlss()`

function.

### Note

The response variable should be a matrix containing two columns, the first with the count of successes and the second with the count of failures.
The parameter `mu`

represents a probability parameter with limits *0 < mu <1*.
*n*mu* is the mean of the distribution where n is the binomial denominator.

### Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

### References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion),
*Appl. Statist.*, **54**, part 3, pp 507-554.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also http://www.gamlss.org/).

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R.
*Journal of Statistical Software*, Vol. **23**, Issue 7, Dec 2007, http://www.jstatsoft.org/v23/i07.

### See Also

`gamlss.family`

, `ZABI`

, `ZIBI`

### Examples

1 2 3 4 5 6 7 8 | ```
BI()# gives information about the default links for the Binomial distribution
# data(aep)
# library(gamlss)
# h<-gamlss(y~ward+loglos+year, family=BI, data=aep)
# plot of the binomial distribution
curve(dBI(x, mu = .5, bd=10), from=0, to=10, n=10+1, type="h")
tN <- table(Ni <- rBI(1000, mu=.2, bd=10))
r <- barplot(tN, col='lightblue')
``` |