Quantile function of the hypergeometric distribution
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stats::hypergeometricQuantile(N
, X
, n
)
stats::hypergeometricQuantile(N, X, n)
returns
a procedure representing the quantile function (discrete inverse)
of the cumulative distribution function stats::hypergeometricCDF(N,
X, n)
. For 0 ≤ x ≤
1, k = stats::hypergeometricQuantile(N, X, n)(x) is
the smallest nonnegative integer satisfying
.
The procedure f:=stats::hypergeometricQuantile(N, X,
n)
can be called in the form f(x)
with
arithmetical expressions x
. The return value of f(x)
is
either a natural number between 0 and min(X, n),
or a symbolic expression:
If x is
a real number satisfying 0 ≤ x ≤
1, 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, then in some cases a symbolic result will be returned:
0 will be returned if either any of x, n or X is zero or if n = 1 and or if X = 1 and .
1 will be returned if n = 1 and or if X = 1 and .
n will be returned if X = N  1 and or if X = N and x > 0.
X will be returned if n = N  1 and or if n = N and x > 0.
n  1 will be returned if X = N  1 and provided that n is symbolic, whereas X  1 will be returned if n = N  1 and provided that X is symbolic.
Finally min(X, n) will be returned if x = 1.
The symbolic call stats::hypergeometricQuantile(N,
X, n)(x)
is returned by f(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.
If x
is a floatingpoint 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: Note that if floatingpoint arguments are passed to the quantile function f, the result is computed with floatingpoint arithmetic. This is faster than using exact arithmetic, but the result is subject to internal roundoff errors. In particular, roundoff may be significant for arguments x close to 1. Cf. Example 4. 
The quantile value k = stats::hypergeometricQuantile(N, X, n)(x) satisfies
,
where cdf = stats::hypergeometricCDF(N, X, n).
The function is sensitive to the environment variable DIGITS
which
determines the numerical working precision.
We evaluate the quantile function with N = 50, X = 30 and n = 10 at some points:
f := stats::hypergeometricQuantile(50, 30, 10): f(0), f((2/3)^30), f(PI/10), f(0.5), f(1  1/10^10)
With cdf = stats::hypergeometricCDF(N, X, n), the quantile value f(x) satisfies the inequalities cdf(f(x)  1) < x ≤ cdf(f(x)):
x := 0.7: f(x)
stats::hypergeometricCDF(50, 30, 10)(float(f(x)  1)), x, stats::hypergeometricCDF(50, 30, 10)(float(f(x)))
delete f, x:
We use symbolic arguments:
f := stats::hypergeometricQuantile(N, X, n): f(x), f(9/10)
When N
, X
and n
evaluate
to suitable numbers, the function f
starts to produce
quantile values:
N := 500: X := 80: n := 18: f(1/2), f(999/1000), f(1  1/10^10), f(1  1/10^80)
delete f, N, X, n:
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::hypergeometricQuantile(N, X, n): f(0), f(1)
X := N  1: f(n/N), f(7/10)
assume(x > n/N): f(0.5), f(x)
delete f, X, x:
If floatingpoint arguments are passed to the quantile function, the result is computed with floatingpoint arithmetic. This is faster than using exact arithmetic, but the result is subject to internal roundoff errors:
f := stats::hypergeometricQuantile(10000, 2000, 30): f(1  1/10^18) <> f(float(1  1/10^18))
delete f:

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

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

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