![]() | ||
Parameters N ∈ { 0 , 1 , 2 , … } K ∈ { 0 , 1 , 2 , … , N } n ∈ { 0 , 1 , 2 , … , N } {displaystyle {egin{aligned}N&in left{0,1,2,dots ight}K&in left{0,1,2,dots ,Night}&in left{0,1,2,dots ,Night}end{aligned}},} Support k ∈ { max ( 0 , n + K − N ) , … , min ( n , K ) } {displaystyle scriptstyle {k,in ,left{max {(0,,n+K-N)},,dots ,,min {(n,,K)}ight}},} pmf ( K k ) ( N − K n − k ) ( N n ) {displaystyle {{{K choose k}{{N-K} choose {n-k}}} over {N choose n}}} CDF 1 − ( n k + 1 ) ( N − n K − k − 1 ) ( N K ) 3 F 2 [ 1 , k + 1 − K , k + 1 − n k + 2 , N + k + 2 − K − n ; 1 ] , {displaystyle 1-{{{n choose {k+1}}{{N-n} choose {K-k-1}}} over {N choose K}},_{3}F_{2}!!left[{egin{array}{c}1, k+1-K, k+1-nk+2, N+k+2-K-nend{array}};1ight],} where p F q {displaystyle ,_{p}F_{q}} is the generalized hypergeometric function Mean n K N {displaystyle n{K over N}} Mode ⌊ ( n + 1 ) ( K + 1 ) N + 2 ⌋ {displaystyle leftlfloor {rac {(n+1)(K+1)}{N+2}}ightfloor } |
In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of
Contents
- Definition
- Combinatorial identities
- Application and example
- Application to Texas Holdem Poker
- Symmetries
- Hypergeometric test
- Relationship to Fishers exact test
- Order of draws
- Related distributions
- Tail bounds
- Multivariate hypergeometric distribution
- Example
- References
In statistics, the hypergeometric test uses the hypergeometric distribution to calculate the statistical significance of having drawn a specific
Definition
The following conditions characterize the hypergeometric distribution:
A random variable
where
The pmf is positive when
The pmf satisfies the recurrence relation
with
Combinatorial identities
As one would expect, the probabilities sum up to 1:
This is essentially Vandermonde's identity from combinatorics.
Also note the following identity holds:
This follows from the symmetry of the problem, but it can also be shown by expressing the binomial coefficients in terms of factorials and rearranging the latter.
Application and example
The classical application of the hypergeometric distribution is sampling without replacement. Think of an urn with two types of marbles, red ones and green ones. Define drawing a green marble as a success and drawing a red marble as a failure (analogous to the binomial distribution). If the variable N describes the number of all marbles in the urn (see contingency table below) and K describes the number of green marbles, then N − K corresponds to the number of red marbles. In this example, X is the random variable whose outcome is k, the number of green marbles actually drawn in the experiment. This situation is illustrated by the following contingency table:
Now, assume (for example) that there are 5 green and 45 red marbles in the urn. Standing next to the urn, you close your eyes and draw 10 marbles without replacement. What is the probability that exactly 4 of the 10 are green? Note that although we are looking at success/failure, the data are not accurately modeled by the binomial distribution, because the probability of success on each trial is not the same, as the size of the remaining population changes as we remove each marble.
This problem is summarized by the following contingency table:
The probability of drawing exactly k green marbles can be calculated by the formula
Hence, in this example calculate
Intuitively we would expect it to be even more unlikely for all 5 marbles to be green.
As expected, the probability of drawing 5 green marbles is roughly 35 times less likely than that of drawing 4.
Application to Texas Hold'em Poker
In Hold'em Poker players make the best hand they can combining the two cards in their hand with the 5 cards (community cards) eventually turned up on the table. The deck has 52 and there are 13 of each suit. For this example assume a player has 2 clubs in the hand and there are 3 cards showing on the table, 2 of which are also clubs. The player would like to know the probability of one of the next 2 cards to be shown being a club to complete the flush.
(Note that the odds calculated in this example assume no information is known about the cards in the other players' hands; however, experienced poker players may consider how the other players place their bets (check, call, raise, or fold) in considering the odds for each scenario. Strictly speaking, the approach to calculating success probabilities outlined here is accurate in a scenario where there is just one player at the table; in a multiplayer game these odds might be adjusted somewhat based on the betting play of the opponents.)
There are 4 clubs showing so there are 9 still unseen. There are 5 cards showing (2 in the hand and 3 on the table) so there are
The probability that one of the next two cards turned is a club can be calculated using hypergeometric with
The probability that both of the next two cards turned are clubs can be calculated using hypergeometric with
The probability that neither of the next two cards turned are clubs can be calculated using hypergeometric with
Symmetries
Swapping the roles of green and red marbles:
Swapping the roles of drawn and not drawn marbles:
Swapping the roles of green and drawn marbles:
Hypergeometric test
The hypergeometric test uses the hypergeometric distribution to measure the statistical significance of having drawn a sample consisting of a specific number of
Relationship to Fisher's exact test
The test based on the hypergeometric distribution (hypergeometric test) is identical to the corresponding one-tailed version of Fisher's exact test ). Reciprocally, the p-value of a two-sided Fisher's exact test can be calculated as the sum of two appropriate hypergeometric tests (for more information see ).
Order of draws
The probability of drawing any sequence of white and black marbles (the hypergeometric distribution) depends only on the number of white and black marbles, not on the order in which they appear; i.e., it is an exchangeable distribution. As a result, the probability of drawing a white marble in the
Related distributions
Let X ~ Hypergeometric(
where
The following table describes four distributions related to the number of successes in a sequence of draws:
Tail bounds
Let X ~ Hypergeometric(
Where
is the Kullback-Leibler divergence and it is used that
If n is larger than N/2, it can be useful to apply symmetry to "invert" the bounds, which give you the following:
Multivariate hypergeometric distribution
The model of an urn with black and white marbles can be extended to the case where there are more than two colors of marbles. If there are Ki marbles of color i in the urn and you take n marbles at random without replacement, then the number of marbles of each color in the sample (k1,k2,...,kc) has the multivariate hypergeometric distribution. This has the same relationship to the multinomial distribution that the hypergeometric distribution has to the binomial distribution—the multinomial distribution is the "with-replacement" distribution and the multivariate hypergeometric is the "without-replacement" distribution.
The properties of this distribution are given in the adjacent table, where c is the number of different colors and
Example
Suppose there are 5 black, 10 white, and 15 red marbles in an urn. You reach in and randomly select six marbles without replacement. What is the probability that you pick exactly two of each color?
Note: When picking the six marbles with replacement, the expected number of black marbles is 6×(5/30) = 1, the expected number of white marbles is 6×(10/30) = 2, and the expected number of red marbles is 6×(15/30) = 3. This comes from the expected value of a Binomial distribution, E(X) = np.