# Documentation

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# `stats`::`betaCDF`

Cumulative distribution function of the beta distribution

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## Syntax

```stats::betaCDF(`a`, `b`)
```

## Description

`stats::betaCDF(a, b)` returns a procedure representing the cumulative distribution function

.

of the beta distribution with shape parameters a > 0, b > 0.

The procedure `f := stats::betaCDF(a, b)` can be called in the form `f(x)` with an arithmetical expression `x`. The return value of `f(x)` is either a floating-point number or a symbolic expression:

• If `x` can be converted to a real floating-point number and `a` and `b` can be converted to positive floating-point numbers, then the return value `f(x)` is a floating-point number.

• For numerical values x ≤ 0 and x ≥ 1, the floating-point numbers `0.0`, respectively `1.0`, are returned even if `a` and `b` are symbolic quantities.

• The call `f(- infinity )` returns `0.0`; the call `f( infinity )` return `1.0`.

• In all other cases, `f(x)` returns the symbolic call `stats::betaCDF(a, b)(x)`.

Numerical values of `a` and `b` are only accepted if they are positive.

## Environment Interactions

The function is sensitive to the environment variable `DIGITS` which determines the numerical working precision.

## Examples

### Example 1

We evaluate the cumulative distribution function with a = 5 and b = 7 at various points:

```f := stats::betaCDF(5, 7): f(-infinity), f(-PI), f(1/sqrt(10)), f(0.75), f(1), f(infinity)```

Nonpositive numerical values of a or b lead to an error:

`stats::betaCDF(-5, 7)(0.75)`
```Error: First shape parameter must be positive. [stats::betaCDF] ```
```Error: the first shape parameter must be positive [stats::betaCDF] ```
`delete f:`

### Example 2

For symbolic arguments, symbolic calls of `stats::betaCDF` are returned, unless x ≤ 0 or x ≥ 1 can be decided:

```f := stats::betaCDF(a, b): f(-2), f(0), f(1/3), f(0.4), f(1), f(PI), f(x)```

When positive real numbers are assigned to a and b, the call f(x) returns a floating-point number if x is numerical:

```a := 2: b := PI: f(-2), f(1/3), f(0.4), f(PI)```

`delete f, a, b:`

## Parameters

 `a`, `b` The shape parameters of the beta distribution: arithmetical expressions representing positive real values.