# Documentation

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# `stats`::`chisquareCDF`

Cumulative distribution function of the chi-square distribution

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## Syntax

```stats::chisquareCDF(`m`)
```

## Description

`stats::chisquareCDF(m)` returns a procedure representing the cumulative distribution function

of the chi-square distribution with mean m > 0.

The procedure `f := stats::chisquareCDF(m)` 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 ≤ 0 can be decided, then `f(x)` returns 0. If x > 0 can be decided, then `f(x)` returns the value .

If x is a floating-point number and m can be converted to a positive floating-point number, then these values are returned as floating-point numbers. Otherwise, symbolic expressions are returned.

The function `f` reacts to properties of identifiers set via `assume`. If x is a symbolic expression with the property x ≤ 0 or x ≥ 0, the corresponding values are returned.

`f(x)` returns the symbolic call `stats::chisquareCDF(m)(x)`if neither x ≤ 0 nor x > 0 can be decided.

Numerical values for `m` are only accepted if they are real and positive.

Note that, for large `m`, exact results may be costly to compute. If floating-point values are desired, it is recommended to pass floating-point arguments `x` to `f` rather than to compute exact results `f(x)` and convert them via `float`. Cf. Example 4.

## 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 mean m = 2 at various points:

```f := stats::chisquareCDF(2): f(-infinity), f(-3), f(1/2), f(0.5), f(PI), f(infinity)```

`delete f:`

### Example 2

If `x` is a symbolic object without properties, then it cannot be decided whether x ≥ 0 holds. A symbolic function call is returned:

```f := stats::chisquareCDF(m): f(x)```

With suitable properties, it can be decided whether x ≥ 0 holds. An explicit expression is returned:

```assume(0 <= x): f(x)```

For integer values of `m`, the special function `igamma` can be expressed in terms of more elementary functions:

```m := 6: f(x)```

```m := 5: f(x)```

`unassume(x): delete f, m:`

### Example 3

We use a symbolic mean `m`:

```f := stats::chisquareCDF(m): f(3), f(3.0)```

When a numerical value is assigned to `m`, the function `f` starts to produce numerical values:

```m := PI: f(3), f(3.0)```

`delete f, m:`

### Example 4

We consider a chi-square distribution with large mean `m` = 1000:

`f := stats::chisquareCDF(1000):`

For floating-point approximations, one should not compute an exact result and convert it via `float`. For large mean m, it is faster to pass a floating-point argument to `f`. The following call takes some time, because an exact computation of the huge integer `gamma(m/2) = gamma(500) = 499!` is involved:

`float(f(1023))`

The following call is much faster:

`f(float(1023))`

`delete f:`

## Parameters

 `m` The mean: an arithmetical expression representing a positive real value