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# betainc

Evaluates the incomplete beta function.

## Prototype

```	function betainc (
x  : numeric,
a  : numeric,
b  : numeric
)

return_val [dimsizes(x)] :  typeof(x)
```

## Arguments

x

upper limit of integration. x may be of any dimensionality. x must be in (0,1) inclusive and can only be float or double.

a

first beta distribution parameter; must be > 0.0. It must be the same dimensionality as x.

b

second beta distribution parameter; must be > 0.0. It must be the same dimensionality as x.

## Return value

The variable returned will be the same type and dimensionality as x.

As of NCL version 4.3.1, if x contains missing values, the return value will contain missing values in the same locations.

## Description

betainc calculates the incomplete beta function. The incomplete beta function ratio is the probability that a random variable from a beta distribution having parameters a and b will be less than or equal to x. The code used is from SLATEC (http://www.netlib.org/slatec/fnlib/). This returns the same answers as the Numerical Recipes [Cambridge Univ. Press, 1986] function betai.

This function is often used to determine probabilities.

Note: in NCL version 4.3.1, this function was updated to handle missing values. If any missing values are inputted, the output array will contain missing values in the same locations.

## Examples

Example 1

```
a = 0.5
b = 5.0
x = 0.2

alpha = betainc(x,a,b)
print("alpha(x,a,b)="+alpha)

x = 0.5
alpha = betainc(x,a,b)
print("alpha(x,a,b)="+alpha)
```
The result is:
```  alpha(x,a,b)= 0.85507
alpha(x,a,b)= 0.98988
```
Example 2 - The betainc can be used as a p-Value calculator for the Student t-test. Let's say a calculation has been made where the degrees-of-freedom (df=20) and a Student-t value of 2.08 has been determined. A probability level may be determined via:
```
df   = 20
tval = 2.08
prob = betainc( df/(df+tval^2), df/2.0, 0.5)
print ("prob="+prob)
```
The result is prob = 0.0506. This is a two-tailed probability. The one-tailed probability is 0.5*prob = 0.0253,

For plotting, users often prefer to plot the quantity:

```   prob = (1.-betainc(x,a,b))*100.  ; probability in %
```