# BETAINCINV

## Purpose:

Calculates the inverse regularized incomplete beta function.

## Syntax:

BETAINCINV(y, a, b, "tail")

y

-

A series or real number where 0 <= y <= 1.

a

-

A series or real number > 0.

b

-

A series or real number > 0.

"tail"

-

Optional. A string, the upper or lower function flag.

 "lower": lower inverse incomplete beta function (default) "upper": upper inverse incomplete beta function

## Returns:

A scalar or series.

## Example:

y = betainc(.3, .4, .5)

x = betaincinv(y, .4, .5)

x == 0.3.

## Example:

W1: betainc({0.0, 0.5, 1.0}, 0.1, 0.2)

W2: betaincinv(W1)

W1 == {0.0, 0.5, 1.0}.

## Example:

betaincinv(0..0.01..1, 8, 10);

xlabel("y");ylabel("I_x^{\;-1}(8,10)");label("Inverse Incomplete Beta Function")

returns a series consisting of the inverse regularized incomplete beta function for 101 values of y between 0 and 1, with a = 8 and b = 10.

## Remarks:

The beta function is defined as:

The beta function is related to the gamma function by:

The incomplete beta function is a generalization of the beta function and defined as:

BETAINC computes the regularized incomplete beta function:

BETAINCINV computes the inverse of BETAINC.

"Tail" specifies the upper or lower inverse incomplete beta function where "lower" computes the inverse of the integral from 0 to x and "upper" computes the inverse of the integral from x to 1.

betaincinv(1 - y, a, b) = betaincinv(y, a, b, "upper")

If the inputs y, a and b are series, they must all be of the same size.