InverseBetaRegularized

InverseBetaRegularized[s,a,b]

gives the inverse of the regularized incomplete beta function.

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
With the regularized incomplete beta function defined by , InverseBetaRegularized[s,a,b] is the solution for z in .
InverseBetaRegularized[z₀,s,a,b] gives the inverse of BetaRegularized[z₀,z,a,b].
Note that the arguments of InverseBetaRegularized are arranged differently than in InverseGammaRegularized.
For certain special arguments, InverseBetaRegularized automatically evaluates to exact values.
InverseBetaRegularized can be evaluated to arbitrary numerical precision.
InverseBetaRegularized automatically threads over lists.

Examples

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Basic Examples (2)

Evaluate numerically:

Plot over a subset of the reals:

Scope (16)

Numerical Evaluation (3)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

Specific Values (4)

Values of InverseBetaRegularized at fixed points:

Values at zero:

Find a value of z for which the InverseBetaRegularized[z,1,2]=0.5:

TraditionalForm formatting:

Visualization (2)

Plot the InverseBetaRegularized function for different values of parameter a:

Plot the InverseBetaRegularized function for different values of parameter b:

Function Properties (5)

is analytic on the open interval :

It has both singularities and discontinuities at the endpoints 0 or at 1:

is non-negative on the unit interval:

is injective:

is nondecreasing on the unit interval:

is neither convex nor concave on the unit interval:

Differentiation (2)

First derivative with respect to s when a=2 and b=3:

First derivative with respect to a when b=2:

First derivative with respect to b when a=2:

Higher derivatives with respect to s when a=2 and b=3:

Plot the higher derivatives with respect to s when a=2 and b=3:

Generalizations & Extensions (2)

InverseBetaRegularized threads elementwise over lists:

Evaluate the 4-argument generalized case:

Applications (2)

Model the PDF of the beta distribution through uniformly distributed random numbers:

Compare binned modeled distribution with exact distribution:

A multivariate Student copula:

Probability density function:

Properties & Relations (2)

InverseBetaRegularized is the inverse of BetaRegularized:

Solve a transcendental equation:

Possible Issues (2)

InverseBetaRegularized evaluates numerically only for :

In TraditionalForm, is not automatically interpreted as an inverse regularized beta function:

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