dhyper(q, m, n, k, log = FALSE) phyper(q, m, n, k) qhyper(p, m, n, k) rhyper(nn, m, n, k, bigdata=F)
bdVector
of values of a random variable representing the number of red balls
out of a sample of size
k
drawn from an urn containing
m
red balls and
n
black ones.
bdVector
of probabilities. Missing values (
NA
s) are allowed. Its values should be between 0 and 1.
nn
random hypergeometrically distributed numbers are returned
unless
length(nn)
is larger than 1, in which case
length(nn)
random numbers
are returned.
bdVector
with non-negative integer elements.
bdVector
with non-negative integer elements.
m
red and
n
black balls. This can be a vector or
bdVector
like
m
and
n
.
TRUE
, an object of type
bdVector
is returned.
Otherwise, a
vector
object is returned. This argument can be used only if the bigdata library section has been loaded.
TRUE
, dhyper will return the log of the
density, not the density itself.
dhyper
returns discrete probability values.
Other functions return vectors or
bdVector
s of cumulative probabilities (
phyper
), quantiles
(
qhyper
), or random samples (
rhyper
) for the Hypergeometric distribution.
rhyper
causes creation of the dataset
.Random.seed
if it does
not already exist, otherwise its value is updated.
Missing values (
NA
s) and
+-Inf
s
are allowed as components of
q
,
p
or
nn
,
but not in the vectors or
bdVector
s of parameters.
If
q
,
m
,
n
, or
k
are vectors or
bdVector
s of different lengths, each is
replicated cyclically to the length of
the longest.
The values of
q
,
m
,
n
, and
k
are rounded to the nearest integer value before any calculations are made.
The Hypergeometric distribution can be described
by an Urn Model with
m
red and
n
black balls.
Any sequence of
k
drawings resulting in
k-q
black and
q
red balls
has the same probability. It is similar to the Binomial distribution
but sampled from a finite population without replacement.
A hypergeometric variable corresponds to the conditional distribution
of the number in
the upper left cell of a 2 by 2 table with row marginal totals
m
and
n
and column marginal totals
k
and
N-k
,
if the unconditional distributions of cell counts are
Poisson,
where
N=m+n
is the grand total.
By symmetry between rows and columns,
phyper(q, m, n, k) = phyper(q, k, N-k, m)
.
The range of the distribution is
max(0, k-n) <= q <= min(m, k)
,
the density is
p(q, m, n, k) = choose(m, q) * choose(n, k-q) / choose(N, k)
,
the expected value is
m * k / N
,
and variance is
m * n * k * (N-k) / (N^2 * (N-1))
.
For details on the uniform random number generator implemented in S-PLUS,
see the
set.seed
help file.
Hoel, P., Port, S. and Stone, C. (1971).
Introduction to Probability Theory. Houghton-Mifflin, Boston, MA.
Johnson, N. L. and Kotz, S. (1970).
Discrete Univariate Distributions, vol. 2.
Houghton-Mifflin, Boston, MA.
cumsum(dhyper(0:5,4,6,7)) # cumulative distribution function phyper(0:5,4,6,7) # same thing phyper(0:5,7,3,4) # same thing, by symmetry of rows and columns rhyper(10,4,6,7) # 10 random values dhyper(rep(3,3), m=c(5,8,12), n=4, k=4)