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# stats::binomialCDF

The (discrete) cumulative distribution function of the binomial distribution

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## 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.