Greisen–Zatsepin–Kuzmin limit: Difference between revisions

Revision as of 03:39, 17 November 2013

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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 $X$ follows the hypergeometric distribution if its probability mass function (pmf) is given by:^[1]

P (X = k) = \frac{(\binom{K}{k}) (\binom{N - K}{n - k})}{(\binom{N}{n})}

Where:

$N$ is the population size
$K$ is the number of success states in the population
$n$ is the number of draws
$k$ is the number of successes
$(\binom{a}{b})$ is a binomial coefficient

The pmf is positive when $\max (0, n + K - N) \leq k \leq \min (K, n) .$

Combinatorial identities

As one would expect, the probabilities sum up to 1 :

$\sum_{0 \leq k \leq K} \frac{(\binom{K}{k}) (\binom{N - K}{n - k})}{(\binom{N}{n})} = 1$

This is essentially Vandermonde's identity from combinatorics.

Also note the following identity holds:

\frac{(\binom{K}{k}) (\binom{N - K}{n - k})}{(\binom{N}{n})} = \frac{(\binom{n}{k}) (\binom{N - n}{K - k})}{(\binom{N}{K})} .

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, 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:

	drawn	not drawn	total
white marbles	k	K − k	K
black marbles	n − k	N + k − n − K	N − K
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:

	drawn	not drawn	total
white marbles	k = 4	K − k = 1	K = 5
black marbles	n − k = 6	N + k − n − K = 39	N − K = 45
total	n = 10	N − n = 40	N = 50

The probability of drawing exactly k white marbles can be calculated by the formula

P (X = k) = f (k; N, K, n) = \frac{(\binom{K}{k}) (\binom{N - K}{n - k})}{(\binom{N}{n})} .

Hence, in this example calculate

P (X = 4) = f (4; 50, 5, 10) = \frac{(\binom{5}{4}) (\binom{45}{6})}{(\binom{50}{10})} = \frac{5 \cdot 8145060}{10272278170} = 0.003964583 \dots .

Intuitively we would expect it to be even more unlikely for all 5 marbles to be white.

P (X = 5) = f (5; 50, 5, 10) = \frac{(\binom{5}{5}) (\binom{45}{5})}{(\binom{50}{10})} = \frac{1 \cdot 1221759}{10272278170} = 0.0001189375 \dots,

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.

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:

f (k; N, K, n) = f (n - k; N, N - K, n)

Swapping the roles of drawn and not drawn marbles:

f (k; N, K, n) = f (K - k; N, K, N - n)

Swapping the roles of white and drawn marbles:

f (k; N, K, n) = f (k; N, n, K)

Relationship to Fisher's exact test

The test (see aboveTemplate:Clarify) based on the hypergeometric distribution (hypergeometric test) is identical to the corresponding one-tailed version of Fisher's exact test^[2] ). 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^[3] ).

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 $i^{th}$ draw isPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

P (W_{i}) = \frac{K}{N} .

Related distributions

Let X ~ Hypergeometric( $K$ , $N$ , $n$ ) and $p = K / N$ .

If $n = 1$ then $X$ has a Bernoulli distribution with parameter $p$ .

Let $Y$ have a binomial distribution with parameters $n$ and $p$ ; this models the number of successes in the analogous sampling problem with replacement. If $N$ and $K$ are large compared to $n$ and $p$ is not close to 0 or 1, then $X$ and $Y$ have similar distributions, i.e., $P (X \leq k) \approx P (Y \leq k)$ .

If $n$ is large, $N$ and $K$ are large compared to $n$ and $p$ is not close to 0 or 1, then

P (X \leq k) \approx Φ (\frac{k - n p}{\sqrt{n p (1 - p)}})

where $Φ$ is the 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 $X$ has a noncentral hypergeometric distribution

Multivariate hypergeometric distribution

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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 K_i 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,...,k_c) 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 $N = \sum_{i = 1}^{c} K_{i}$ 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?

P (2 black, 2 white, 2 red) = \frac{(\binom{5}{2}) (\binom{10}{2}) (\binom{15}{2})}{(\binom{30}{6})} = . 079575596816976

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.

Notes

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↑ Template:Cite web

References

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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

Template:Cite web unpublished note

External links

The Hypergeometric Distribution and Binomial Approximation to a Hypergeometric Random Variable by Chris Boucher, Wolfram Demonstrations Project.
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[1] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[2] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[3] Template:Cite web

[1]

[2]

[3]

Greisen–Zatsepin–Kuzmin limit: Difference between revisions

Revision as of 03:39, 17 November 2013

Contents

Definition

Combinatorial identities

Application and example

Application to Texas Hold'em Poker

Symmetries

Relationship to Fisher's exact test

Order of draws

Related distributions

Multivariate hypergeometric distribution

Example

See also

Notes

References

External links

Navigation menu

@@ Line 1: / Line 1: @@
+{{Expert-subject|Statistics|reason=over-verbose, but no proper mention of "hypergeometric test", which redirects here|date=April 2013}}
+<!-- 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.
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+==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''&nbsp;−&nbsp;''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]]

Greisen–Zatsepin–Kuzmin limit: Difference between revisions

Revision as of 03:39, 17 November 2013

Definition

Combinatorial identities

Application and example

Application to Texas Hold'em Poker

Symmetries

Relationship to Fisher's exact test

Order of draws

Related distributions

Multivariate hypergeometric distribution

Example

See also

Notes

References

External links

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