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stats::hypergeometricCDF

The (discrete) cumulative probability function of the hypergeometric distribution

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Syntax

stats::hypergeometricCDF(N, X, n)

Description

stats::hypergeometricCDF(N, X, n) returns a procedure representing the probability function

of the hypergeometric distribution with "population size" N, "success population size" X and "sample size" n.

The procedure f:=stats::hypergeometricCDF(N, X, n) can be called in the form f(x) with arithmetical expressions x. The return value of f(x) is either a floating-point number, an exact numerical value, or a symbolic expression:

If x is an integer, a rational or a floating point number, while N is a positive integer and both X and n are nonnegative integers, then an explicit numerical value is returned.

The function f reacts to properties of identifiers set via assume.

If any of the parameters is symbolic with properties as follows, then 0, 1 or a symbolic result is returned:

If x < max(0, n + X - N), then f(x) = 0.

If xmin(n, X), then f(x) = 1.

If X = N, then f(x) = 0 for x < n and f(x) = 1 for xn.

If n = N, then f(x) = 0 for x < X and f(x) = 1 for xX.

If X = N - 1, then f(x) = 0 for x < n - 1, for n - 1 ≤ x < n and f(x) = 1 for xn.

If n = N - 1, then f(x) = 0 for x < X - 1, for X - 1 ≤ x < X and f(x) = 1 for xX.

If X = 1, then for 0 ≤ x < 1 and f(x) = 1 for x ≥ 1.

If n = 1, then for 0 ≤ x < 1 and f(x) = 1 for x ≥ 1.

If X = 0 or n = 0, then f(x) = 1 for x ≥ 0.

If x and all parameters but N are numerical and the assumption on N is assume(N > X), then symbolic values are returned.

f(x) returns the symbolic call stats::hypergeometricCDF(N, X, n)(x) in all other cases.

Numerical values for N are only accepted if they are positive integers.

Numerical values for X are only accepted if they are nonnegative integers.

Numerical values for n are only accepted if they are nonnegative integers.

    Note:   If x is a floating-point number, the result is a floating number provided N, X and n are numerical values. If x is an exact value, the result is a rational number.

Note that for large numbers, floating-point results are computed much faster than exact results. If floating-point approximations are desired, pass a floating-point number x to stats::hypergeometricCDF.

Environment Interactions

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

Examples

Example 1

We compute the distribution function with N = 20, X = 4 and n = 3 at various points:

f := stats::hypergeometricCDF(20, 4, 3): 
f(-1), f(0), f(1/2), f(1), f(2), f(PI), f(5)

f(-infinity), f(infinity)

f(-0.2), f(0.0), f(0.7), f(1.0), f(float(PI)), f(4.0)

delete f:

Example 2

We use symbolic arguments:

f := stats::hypergeometricCDF(N, X, n): f(x), f(8), f(8.0)

When real numbers are assigned to N, X and n, the function f starts to produce explicit results if the argument is numerical:

N := 15: X := 6: n := 5:
f(0), f(1), f(2.0), f(3.5), f(4)

delete f, N, X, n:

Example 3

If one or more parameters are symbolic, usually a symbolic call is returned. Some combinations of symbolic and numeric values for N, X, n and x, however, may yield symbolic or numeric results:

f := stats::hypergeometricCDF(N, X, n):
X := 1:
f(-1), f(0), f(1/2), f(0.5), f(3/2), f(2.0)

X := N - 1:
f(1), f(n-1), f(n)

delete X:

Example 4

If x and all parameters but N are numerical and N is assumed to be greater than X, a symbolic expression is returned:

X := 6:
assume(N > X):
f := stats::hypergeometricCDF(N, X, 5):
f(1), f(2), f(3)

delete f, N, X:

Parameters

N

The "population size": an arithmetical expression representing a positive integer

X

The "success population size": an arithmetical expression representing a nonnegative integer

n

The "sample size": an arithmetical expression representing a nonnegative integer

Return Values

procedure.

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