# Documentation

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# `stats`::`binomialCDF`

The (discrete) cumulative distribution function of the binomial distribution

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## Syntax

```stats::binomialCDF(`n`, `p`)
```

## Description

`stats::binomialCDF(n, p)` returns a procedure representing the (discrete) cumulative distribution function

.

of the binomial distribution with “trial parameter” `n` and “probability parameter” `p`.

The procedure `f := stats::binomialCDF(n, p)` 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, an exact numerical value, or a symbolic expression:

• If `x` is a numerical real value and `n` is a positive integer, then an explicit value is returned. If `p` is a numerical value satisfying 0 ≤ pp ≤ 1, this is a numerical value. Otherwise, it is a symbolic expression in `p`.

• If `x` is a numerical value with x < 0, then `0`, respectively `0.0`, is returned for any value of `n` and `p`.

• For symbolic values of `n`, explicit results are returned if x is a numerical value with x < 2.

• For symbolic values of `n`, explicit results are returned if n - x is a numerical value with n - x ≤ 2.

• If `n - x` is a numerical value with n - x ≤ 0, then `1`, respectively `1.0`, is returned for any value of `n` and `p`.

• In all other cases, `f(x)` returns the symbolic call `binomialCDF(n, p)(x)`.

Numerical values for `n` are only accepted if they are positive integers.

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

If `x` is a real floating-point number, the result is a floating number provided n and p are numerical values. If `x` is an exact numerical value, the result is an exact number.

### Note

Note that for large n, floating-point results are computed much faster than exact results. If floating-point approximations are desired, pass a floating-point number `x` to the procedure generated by `stats::binomialCDF`!

## 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 n = 20 and at various points:

```f := stats::binomialCDF(5, 3/4): f(-1), f(2), f(PI), f(5), f(6)```

`f(-1.2), f(2.0), f(float(PI)), f(5.5)`

`delete f:`

### Example 2

We use symbolic arguments:

`f := stats::binomialCDF(n, p): f(x), f(8), f(8.0)`

When numerical values are assigned to n and p, the function f starts to produce explicit results if the argument is numerical:

```n := 3: p := 1/3: f(2), f(2.5), f(PI +1), f(4.0)```

`delete f, n, p:`

### Example 3

If n and x are numerical, symbolic expressions are returned for symbolic values of p:

```f := stats::binomialCDF(3, p): f(-1), f(0), f(3/2), f(1 + sqrt(3)), f(2.999), f(3)```

`delete f:`

## Parameters

 `n` The “trial parameter”: an arithmetical expression representing a positive integer `p` The “probability parameter”: an arithmetical expression representing a real number 0 ≤ p ≤ 1.