|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Expert-subject|Statistics|reason=over-verbose, but no proper mention of "hypergeometric test", which redirects here|date=April 2013}}
| | It is really painful to have hemorrhoids. This can be a big hindrance for a daily routine. You cannot do all of the points which we utilize to do due to the pain that you are going by.<br><br>If you have decided which lotions are the [http://hemorrhoidtreatmentfix.com/external-hemorrhoid-treatment external hemorrhoids treatment] you'd like then you would find which they are found inside numerous drug stores plus pharmacies, plus which are reasonably inexpensive.<br><br>If the expressing the most extreme form of hemorrhoids plus it is actually causing you pain, it is very important to find a physician, as the situation will get worse should you do not.<br><br>If you will ask wellness practitioners they usually suggest operation inside getting rid of your hemorrhoid. The procedures in surgery are reasonably simple plus to get out o the painful condition when you are done treating it.<br><br>If you don't already learn what a sitz bathtub is, it is actually merely taking a bathtub inside a sitting down position. When taking these baths, I want you to ensure the water is as warm because you are able to handle as this may relax both we plus the hemorrhoid. It can also allow more blood flow, and blood carries vitamins and minerals which might aid cure it faster.<br><br>Ice is regarded as the simplest yet the most effective hemorrhoid treatments you are able to employ to reduce swelling, inflammation, bleeding and pain. Wrap it in chipped shape in a piece of cheese cloth plus apply it onto the hemorrhoid itself.<br><br>Undergoing with these options can surely expense you expensive. And for certain not all people can afford to pay such surgery. Now there are also hemorrhoids treatment which can be found at home. With these hemorrhoid treatments you are able to be sure that you'll not spend too much. In most situations, folks prefer to have all-natural treatment whilst the hemorrhoid remains on its mild stage. These natural treatments normally enable we in reducing the pain plus swelling. We never have to be concerned as we apply or employ them considering they are fairly simple and affordable. |
| | |
| <!-- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion
| |
| of standards used for probability distribution articles such as this one. -->
| |
| {{Infobox probability distribution
| |
| | name = Hypergeometric
| |
| | type = mass
| |
| | pdf_image =
| |
| | cdf_image =
| |
| | parameters = <math>\begin{align}N&\in \left\{0,1,2,\dots\right\} \\
| |
| K&\in \left\{0,1,2,\dots,N\right\} \\
| |
| n&\in \left\{0,1,2,\dots,N\right\}\end{align}\,</math>
| |
| | support = <math>\scriptstyle{k\, \in\, \left\{\max{(0,\, n+K-N)},\, \dots,\, \min{(K,\, n )}\right\}}\,</math>
| |
| | pdf = <math>{{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}</math>
| |
| | cdf = <math>1-{{{n \choose {k+1}}{{N-n} \choose {K-k-1}}}\over {N \choose K}} \,_3F_2\!\!\left[\begin{array}{c}1,\ k+1-K,\ k+1-n \\ k+2,\ N+k+2-K-n\end{array};1\right]</math>
| |
| | mean = <math>n {K\over N}</math>
| |
| | median =
| |
| | mode = <math>\left \lfloor \frac{(n+1)(K+1)}{N+2} \right \rfloor</math>
| |
| | variance = <math>n{K\over N}{(N-K)\over N}{N-n\over N-1}</math>
| |
| | skewness = <math>\frac{(N-2K)(N-1)^\frac{1}{2}(N-2n)}{[nK(N-K)(N-n)]^\frac{1}{2}(N-2)}</math>
| |
| | kurtosis = <math> \left.\frac{1}{n K(N-K)(N-n)(N-2)(N-3)}\cdot\right.</math>
| |
| <math>\Big[(N-1)N^{2}\Big(N(N+1)-6K(N-K)-6n(N-n)\Big)+</math>
| |
| <math>6 n K (N-K)(N-n)(5N-6)\Big]</math>
| |
| | entropy =
| |
| | mgf = <math>\frac{{N-K \choose n} \scriptstyle{\,_2F_1(-n, -K; N - K - n + 1; e^{t}) } }
| |
| {{N \choose n}} \,\!</math>
| |
| | char = <math>\frac{{N-K \choose n} \scriptstyle{\,_2F_1(-n, -K; N - K - n + 1; e^{it}) }}
| |
| {{N \choose n}} </math>
| |
| }}
| |
| In [[probability theory]] and [[statistics]], the '''hypergeometric distribution''' is a discrete [[probability distribution]] that describes the probability of <math>k</math> successes in <math>n</math> draws ''without'' replacement from a finite [[population]] of size <math>N</math> containing exactly <math>K</math> successes. This is in contrast to the [[binomial distribution]], which describes the probability of <math>k</math> successes in <math>n</math> draws ''with'' replacement.
| |
| | |
| ==Definition==
| |
| | |
| The hypergeometric distribution applies to sampling without replacement from a finite population whose elements can be classified into two mutually exclusive categories like Pass/Fail, Male/Female or Employed/Unemployed. As random selections are made from the population, each subsequent draw decreases the population causing the probability of success to change with each draw.
| |
| | |
| The following conditions characterise the hypergeometric distribution:
| |
| * The result of each draw can be classified into one or two categories.
| |
| * The probability of a success changes on each draw.
| |
| | |
| A [[random variable]] <math>X</math> follows the hypergeometric distribution if its [[probability mass function]] (pmf) is given by:<ref>{{Cite book
| |
| | edition = Third
| |
| | publisher = Duxbury Press
| |
| | last = Rice
| |
| | first = John A.
| |
| | title = Mathematical Statistics and Data Analysis
| |
| | year = 2007
| |
| | page = 42
| |
| }}</ref>
| |
| | |
| :<math> P(X=k) = {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}</math>
| |
| | |
| Where:
| |
| | |
| *<math>N</math> is the population size
| |
| *<math>K</math> is the number of success states in the population
| |
| *<math>n</math> is the number of draws
| |
| *<math>k</math> is the number of successes
| |
| *<math>\textstyle {a \choose b}</math> is a [[binomial coefficient]]
| |
| | |
| The pmf is positive when <math>\max(0, n+K-N) \leq k \leq \min(K,n).</math>
| |
| | |
| ==Combinatorial identities==
| |
| | |
| As one would expect, the probabilities sum up to 1 :
| |
| | |
| <math> \sum_{0\leq k\leq K} { {K \choose k} { N-K \choose n-k} \over {N \choose n} } = 1</math>
| |
| | |
| This is essentially [[Vandermonde's identity]] from [[combinatorics]].
| |
| | |
| Also note the following identity holds:
| |
| | |
| :<math> {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}} = {{{n \choose k} {{N-n} \choose {K-k}}}\over {N \choose K}}.</math>
| |
| | |
| 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 problem|urn]] with two types of [[marbles]], black ones and white ones. Define drawing a white marble as a success and drawing a black 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 '''white marbles''', then ''N'' − ''K'' corresponds to the number of '''black marbles'''. In this example, ''X'' is the [[random variable]] whose outcome is ''k'', the number of white marbles actually drawn in the experiment. This situation is illustrated by the following [[contingency table]]:
| |
| <!-- Formatting problem: tables overlap in Firefox with low resolution unless aligned by right. Please keep align=right!
| |
| {| class="wikitable" style="float:right; margin-left:1em"
| |
| |-
| |
| !
| |
| ! drawn
| |
| ! not drawn
| |
| ! total
| |
| |-
| |
| | align="right" | '''defective'''
| |
| | align="right" | ''k''
| |
| | align="right" | ''K'' − ''k''
| |
| | align="right" | ''K''
| |
| |-
| |
| | align="right" | '''non-defective'''
| |
| | align="right" | ''n'' − ''k''
| |
| | align="right" | ''N − K − n + k''
| |
| | align="right" | ''N − K''
| |
| |-
| |
| | align="right" | '''total'''
| |
| td align="right">''n''
| |
| | align="right" | ''N − n''
| |
| | align="right" | ''N''
| |
| |} {{Clearright}}-->
| |
| {| class="wikitable" style="text-align:center"
| |
| ! || drawn || not drawn || total
| |
| |-
| |
| | align="right" | '''white marbles''' || ''k'' || ''K'' − ''k'' || ''K''
| |
| |-
| |
| | align="right" | '''black marbles''' || ''n'' − ''k'' || ''N + k − n − K'' || ''N − K''
| |
| |-
| |
| | align="right" | '''total''' || ''n'' || ''N − n'' || ''N''
| |
| |-
| |
| |}
| |
| | |
| Now, assume (for example) that there are 5 white and 45 black 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 white? ''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:
| |
| {| class="wikitable" style="text-align:center"
| |
| |-
| |
| ! !! drawn !! not drawn !! total
| |
| |-
| |
| | align="right" | '''white marbles'''
| |
| | ''k'' = '''4'''
| |
| | ''K'' − ''k'' = '''1'''
| |
| | ''K'' = '''5'''
| |
| |-
| |
| | align="right" | '''black marbles'''
| |
| | ''n'' − ''k'' = '''6'''
| |
| | ''N + k − n − K'' = '''39'''
| |
| | ''N − K'' = '''45'''
| |
| |-
| |
| | align="right" | '''total'''
| |
| | ''n'' = '''10'''
| |
| | ''N − n'' = '''40'''
| |
| | ''N'' = '''50'''
| |
| |}
| |
| | |
| The probability of drawing exactly ''k'' white marbles can be calculated by the formula
| |
| | |
| :<math> P(X=k) = f(k;N,K,n) = {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}.</math>
| |
| | |
| Hence, in this example calculate
| |
| | |
| :<math> P(X=4) = f(4;50,5,10) = {{{5 \choose 4} {{45} \choose {6}}}\over {50 \choose 10}} = {5\cdot 8145060\over 10272278170} = 0.003964583\dots. </math>
| |
| | |
| Intuitively we would expect it to be even more unlikely for all 5 marbles to be white.
| |
| | |
| :<math> P(X=5) = f(5;50,5,10) = {{{5 \choose 5} {{45} \choose {5}}}\over {50 \choose 10}} = {1\cdot 1221759
| |
| \over 10272278170} = 0.0001189375\dots, </math>
| |
| | |
| As expected, the probability of drawing 5 white 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 his [[Flush (poker)|flush]].
| |
| | |
| 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 52-5=47 still unseen.
| |
| | |
| The probability that one of the next two cards turned is a club can be calculated using hypergeometric with k=1, n=2, K=9 and N=47.
| |
| | |
| The probability that both of the next two cards turned are clubs can be calculated using hypergeometric with k=2, n=2, K=9 and N=47.
| |
| | |
| The probability that neither of the next two cards turned are clubs can be calculated using hypergeometric with k=0, n=2, K=9 and N=47.
| |
| | |
| == Symmetries ==
| |
| Swapping the roles of black and white marbles:
| |
| : <math> f(k;N,K,n) = f(n-k;N,N-K,n)</math>
| |
| | |
| Swapping the roles of drawn and not drawn marbles:
| |
| : <math> f(k;N,K,n) = f(K-k;N,K,N-n)</math>
| |
| | |
| Swapping the roles of white and drawn marbles:
| |
| : <math> f(k;N,K,n) = f(k;N,n,K) </math>
| |
| | |
| == Relationship to Fisher's exact test ==
| |
| | |
| The test (see above{{clarify|reason=no test mentioned above|date=April 2013}}) based on the hypergeometric distribution (hypergeometric test) is identical to the corresponding one-tailed version of [[Fisher's exact test]]<ref>{{cite journal| first1=I.|last1= Rivals|first2= L. |last2=Personnaz | first3= L. |last3=Taing |first4= M.-C |last4=Potier| title=Enrichment or depletion of a GO category within a class of genes: which test? |volume= 23|journal= Bioinformatics |year=2007 |pages= 401–407|pmid=17182697| doi=10.1093/bioinformatics/btl633| issue=4}}</ref> ). 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<ref>{{cite web| author=K. Preacher and N. Briggs| title=Calculation for Fisher's Exact Test: An interactive calculation tool for Fisher's exact probability test for 2 x 2 tables (interactive page) | url=http://quantpsy.org/fisher/fisher.htm}}</ref> ).
| |
| | |
| == 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 <math>i^{\text{th}}</math> draw is{{citation needed|date=April 2013}}
| |
| | |
| :<math> P(W_i) = {\frac{K}{N}} .</math>
| |
| | |
| == Related distributions ==
| |
| | |
| Let X ~ Hypergeometric(<math>K</math>, <math>N</math>, <math>n</math>) and <math>p=K/N</math>.
| |
| | |
| *If <math>n=1</math> then <math>X</math> has a [[Bernoulli distribution]] with parameter <math>p</math>.
| |
| | |
| *Let <math>Y</math> have a [[binomial distribution]] with parameters <math>n</math> and <math>p</math>; this models the number of successes in the analogous sampling problem ''with'' replacement. If <math>N</math> and <math>K</math> are large compared to <math>n</math> and <math>p</math> is not close to 0 or 1, then <math>X</math> and <math>Y</math> have similar distributions, i.e., <math>P(X \le k) \approx P(Y \le k)</math>.
| |
| | |
| *If <math>n</math> is large, <math>N</math> and <math>K</math> are large compared to <math>n</math> and <math>p</math> is not close to 0 or 1, then
| |
| ::<math>P(X \le k) \approx \Phi \left( \frac{k-n p}{\sqrt{n p (1-p)}} \right)</math>
| |
| | |
| where <math>\Phi</math> is the [[Standard normal distribution#Cumulative distribution function|standard normal distribution function]]
| |
| *If the probabilities to draw a white or black marble are not equal (e.g. because white marbles are bigger/easier to grasp than black marbles) then <math>X</math> has a [[noncentral hypergeometric distribution]]
| |
| | |
| == Multivariate hypergeometric distribution ==
| |
| {{Infobox probability distribution
| |
| | name = Multivariate Hypergeometric Distribution
| |
| | type = mass
| |
| | pdf_image =
| |
| | cdf_image =
| |
| | parameters = <math>c \in \mathbb{N} = \lbrace 0, 1, \ldots \rbrace</math><br /><math>(K_1,\ldots,K_c) \in \mathbb{N}^c</math><br /><math>N = \sum_{i=1}^c K_i</math><br /><math>n \in \lbrace 0,\ldots,N\rbrace</math>
| |
| | support = <math>\left\{ \mathbf{k} \in \mathbb{Z}_{0+}^c \, : \, \forall i\ k_i \le K_i , \sum_{i=1}^{c} k_i = n \right\}</math>
| |
| | pdf = <math>\frac{\prod_{i=1}^{c} \binom{K_i}{k_i}}{\binom{N}{n}}</math>
| |
| | cdf =
| |
| | mean = <math>E(X_i) = \frac{n K_i}{N}</math>
| |
| | median =
| |
| | mode =
| |
| | variance = <math>\text{Var}(X_i) = \frac{K_i}{N} \left(1-\frac{K_i}{N}\right) n \frac{N-n}{N-1} </math><br /><math>\text{Cov}(X_i,X_j) = -\frac{n K_i K_j}{N^2} \frac{N-n}{N-1} </math>
| |
| | skewness =
| |
| | kurtosis =
| |
| | entropy =
| |
| | mgf =
| |
| | char =
| |
| }}
| |
| | |
| The model of an [[urn problem|urn]] with black and white marbles can be extended to the case where there are more than two colors of marbles. If there are ''K''<sub>i</sub> 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 (''k''<sub>1</sub>,''k''<sub>2</sub>,...,''k''<sub>c</sub>) 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 <math>N=\sum_{i=1}^{c} K_i</math> is the total number of marbles.
| |
| | |
| === 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?
| |
| | |
| :<math> P(2\text{ black}, 2\text{ white}, 2\text{ red}) = {{{5 \choose 2}{10 \choose 2} {15 \choose 2}}\over {30 \choose 6}} = .079575596816976</math>
| |
| | |
| ''Note: When picking the six marbles without 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.''
| |
| | |
| <br style="clear:both;" />
| |
| | |
| == See also ==
| |
| * [[Multinomial distribution]]
| |
| * [[Sampling (statistics)]]
| |
| * [[Generalized hypergeometric function]]
| |
| * [[Coupon collector's problem]]
| |
| * [[Geometric distribution]]
| |
| * [[Keno]]
| |
| | |
| {{more footnotes|date=August 2011}}
| |
| | |
| ==Notes==
| |
| <references/>
| |
| | |
| == References ==
| |
| *{{cite journal|doi=10.1016/j.jda.2006.01.001|title=HyperQuick algorithm for discrete hypergeometric distribution|year=2007|last1=Berkopec|first1=Aleš|journal=Journal of Discrete Algorithms|volume=5|issue=2|pages=341}}
| |
| | |
| * {{Cite web|last=Skala|first= M. |year=2011|url=http://ansuz.sooke.bc.ca/professional/hypergeometric.pdf |title=Hypergeometric tail inequalities: ending the insanity}} unpublished note
| |
| | |
| == External links ==
| |
| * [http://demonstrations.wolfram.com/TheHypergeometricDistribution/ The Hypergeometric Distribution] and [http://demonstrations.wolfram.com/BinomialApproximationToAHypergeometricRandomVariable/ Binomial Approximation to a Hypergeometric Random Variable] by Chris Boucher, [[Wolfram Demonstrations Project]].
| |
| * {{MathWorld |title=Hypergeometric Distribution |urlname=HypergeometricDistribution}}
| |
| | |
| {{ProbDistributions|discrete-finite}}
| |
| {{Common univariate probability distributions}}
| |
| | |
| {{DEFAULTSORT:Hypergeometric Distribution}}
| |
| [[Category:Discrete distributions]]
| |
| [[Category:Factorial and binomial topics]]
| |
| [[Category:Probability distributions]]
| |