Accelerating the pace of engineering and science

stats::poissonQuantile

Quantile function of the Poisson distribution

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

```stats::poissonQuantile(m)
```

Description

stats::poissonQuantile(m) returns a procedure representing the quantile function (discrete inverse) of the cumulative distribution function stats::poissonCDF(m). For 0 ≤ x ≤ 1, k = stats::poissonQuantile(m)(x) is the smallest nonnegative integer satisfying

.

The procedure f := stats::poissonQuantile(m) can be called in the form f(x) with an arithmetical expression x. The return value of the call f(x) is either a nonnegative integer, infinity, or a symbolic expression:

If m is a nonnegative real number and x a real number satisfying 0 ≤ x < 1, then f(x) returns a nonnegative integer.

If m = 0, then f(x) returns 0 for any x.

If m ≠ 0, then f(1) and f(1.0) return infinity.

In all other cases, f(x) returns the symbolic call stats::poissonQuantile(m)(x).

Numerical values for m are only accepted if they are positive.

If floating-point arguments are passed to the quantile function f, the result is computed with floating-point arithmetic. This is faster than using exact arithmetic, but the result is subject to internal round-off errors. In particular, round-off may be significant for arguments x close to 1. Cf. Example 3.

Finite quantile values k = stats::poissonQuantile(m)(x) satisfy

.

Environment Interactions

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

Examples

Example 1

We evaluate the quantile function with m = π at various points:

```f := stats::poissonQuantile(PI):
f(0), f(1/20), f(0.3), f(PI/6), f(0.7), f(1-1/10^10), f(1)```

The value f(x) satisfies

:

`x := 0.98: k := f(x)`

```float(stats::poissonCDF(PI)(k - 1)), x,
float(stats::poissonCDF(PI)(k))```

`delete f, x, k:`

Example 2

We use symbolic arguments:

`f := stats::poissonQuantile(m): f(x), f(9/10)`

When m evaluates to a positive real number, the function f starts to produce quantile values:

```m := 17:
f(1/2),  f(999/1000), f(1 - 1/10^10), f(1 - 1/10^80)```

`delete f, m:`

Example 3

If floating-point arguments are passed to the quantile function, the result is computed with floating-point arithmetic. This is faster than using exact arithmetic, but the result is subject to internal round-off errors:

```f := stats::poissonQuantile(123):
f(1 - 1/10^19) <> f(float(1 - 1/10^19))```

`delete f:`

Parameters

 m The mean: a arithmetical expression representing a nonnegative real number