# Documentation

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# `stats`::`geometricCDF`

The (discrete) cumulative distribution function of the geometric distribution

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## Syntax

```stats::geometricCDF(`p`)
```

## Description

`stats::geometricCDF(p)` returns a procedure representing the (discrete) cumulative distribution function

of the geometric distribution with 'probability parameter' `p`.

The procedure `f:=stats::geometricCDF(p)` can be called in the form `f(x)` with an arithmetical expression `x`. The return value of `f(x)` is either a number or a symbolic expression:

If x < 1 can be decided, then `f(x)` returns 0. If x ≥ 1 can be decided, then `f(x)` returns the value .

If x is a floating-point number and p can be converted to a 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 < 1 or x ≥ 1, the corresponding values are returned.

`f(x)` returns the symbolic call `stats::geometricCDF(p)(x)` if neither x < 1 nor x ≥ 1 can be decided.

If p = 0 or p = 0.0, then f(x) returns 0 or 0.0, respectively, for any value of x.

Numerical values for `p` are only accepted if they satisfy 0 ≤ p ≤ 1.

If `x` is a real floating-point number, f(x) produces a floating number provided p is a numerical value. If `x` is an exact numerical value, no internal floating-point conversion of the parameter p is attempted.

## Environment Interactions

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

## Examples

### Example 1

We evaluate the distribution function with at various points:

```f := stats::geometricCDF(1/3): f(-PI) = f(float(-PI)), f(1) = f(1.0), f(103/10) = f(10.3)```

`delete f:`

### Example 2

If `x` is a symbolic object without properties, then it cannot be decided whether x ≥ 1 holds. A symbolic function call is returned:

`f := stats::geometricCDF(p): f(x)`

With suitable properties, it can be decided whether x ≥ 1 holds. An explicit expression is returned:

`assume(1 <= x): f(x)`

`unassume(x): delete f:`

### Example 3

We use symbolic arguments:

`f := stats::geometricCDF(p): f(x)`

If `x` is a numerical value, symbolic expressions in `p` are returned:

`f(-1), f(1), f(5/2), f(PI)`

When numerical values are assigned to p, the function f starts to produce numbers if the argument is numerical:

`p := 1/3: f(-1), f(1), f(5/2), f(PI)`

`delete f, p:`

## Parameters

 `p` The 'probability parameter': an arithmetical expression representing a real number 0 ≤ p ≤ 1.

## Algorithms

The geometric distribution describes the number of Bernoulli trials with success probability p up to and including the first success.