# Documentation

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# `stats`::`exponentialCDF`

Cumulative distribution function of the exponential distribution

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## Syntax

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

## Description

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

of the exponential distribution with real location parameter a and scale parameter b > 0.

The procedure `f := stats::exponentialCDF(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 xa can be decided, then `f(x)` returns 0. If x > a can be decided, then `f(x)` returns the value .

If x is a floating-point number and both a and b can be converted to floating-point numbers, 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 xa or xa, the corresponding values are returned.

`f(x)` returns the symbolic call ```stats::exponentialCDF(a, b)(x)``` if neither xa nor x > a can be decided.

Numerical values for a and b are only accepted if they are real and b is 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 = 0 and b = 1 at various points:

```f := stats::exponentialCDF(0, 1): f(-infinity), f(-PI), f(1/2), f(0.5), f(PI), f(infinity)```

`delete f:`

### Example 2

If `a` or `x` are symbolic objects without properties, then it cannot be decided whether xa holds. A symbolic function call is returned:

`f := stats::exponentialCDF(a, b): f(x)`

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

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

Note that `assume(a <= x)` attached properties both to `a` and `x`. When cleaning up, the properties have to be removed separately for `a` and `x` via `unassume`:

`unassume(a): unassume(x): delete f:`

### Example 3

We use symbolic arguments:

`f := stats::exponentialCDF(a, b): f(x)`

When numerical values are assigned to a and b, the function `f` starts to produce numerical values:

`a := 0: b := 2: f(3), f(3.0)`

`delete f, a, b:`

## Parameters

 `a` The location parameter: an arithmetical expression representing a real value `b` The scale parameter: an arithmetical expression representing a positive real value