Difference between revisions of "HyperGeometric distribution"

Latest revision as of 01:09, 8 December 2018

Release:	4.6 • 5.0 • 5.1 • 5.2 • 5.3 • 5.4 • 6.0 • 6.1 • 6.2 • 6.3 • 6.4 • 6.5 • 6.6

The hypergeometric distribution describes the number of times an event occurs in a fixed number of trials without replacement -- e.g., the number of red balls in a sample of «Trials» balls drawn without replacement from an urn containing «Size» balls of which «PosEvents» are red.

HyperGeometric( 100, 700, 1000 ) →

Functions

HyperGeometric(trials, posEvents, size)

Use this to describe a variable whose outcome has a hyperGeometric distribution.

ProbHyperGeometric(k, trials, posEvents, size)

Returns the probability of outcome «k». It is given by

[math]\displaystyle{ p(k) = { {\binom{posEvents}{k} \binom{size-posEvents}{trials-k} } \over \binom{size}{trials} } }[/math]

Suppose 20 out of 30 balls in a basket are red, and you draw 10 balls at random without replacement. Then the probability of drawing exactly 7 red balls is given by:

ProbHyperGeometric(7, 10, 20, 30) → 0.3096

CumHyperGeometric(k, trials, posEvents, size)

The cumulative probability function for the hyperGeometric distribution. Its value is equal to

[math]\displaystyle{ F(k) = \sum_{i=0}^{k} { {\binom{posEvents}{i} \binom{size-posEvents}{trials-i} } \over \binom{size}{trials} } }[/math]

Use this function when computing the p-Value for a hyperGeometric statistical test.

CumHyperGeometricInv(p, trials, posEvents, size)

The inverse cumulative probability function for the hyperGeometric distribution

Parameters

«trials»: The sample size -— e.g., the number of balls drawn from an urn without replacement. Cannot be larger than «Size».
«posEvents»: The total number of successful events in the population -- e.g, the number of red balls in the urn.
«size»: The population size -- e.g., the total number of balls in the urn, red and non-red.

History

The analytic functions (ProbHyperGeometric, CumHyperGeometric, and CumHyperGeometricInv) were added as built-in functions in Analytica 5.2.
In Analytica 5.1 or earlier, the analytic functions require you to add the Distributions Density Library to your model.

@@ Line 1: / Line 1: @@
-[[Category:Distribution Functions]]
+[[Category:Discrete distributions]]
-[[Category:Doc Status C]] <!-- For Lumina use, do not change -->
+[[Category:Bounded distributions]]
+[[Category:Unimodal distributions]]
+[[Category:Univariate distributions]]
+{{ReleaseBar}}
+The hypergeometric distribution describes the number of times an event occurs in a fixed number of trials without replacement -- e.g., the number of red balls in a sample of «Trials» balls drawn without replacement from an urn containing «Size» balls of which «PosEvents» are red.
-== HyperGeometric(Trials, PosEvents, Size) ==
+<center><code>HyperGeometric( 100, 700, 1000 )</code> &rarr; [[image:hypergeometric_100_700_1000.png]]</center>
-The hypergeometric distribution describes the number of times an event occurs in a fixed number of trials without replacement -- e.g., the number of red balls in a sample of «Trials» balls drawn without replacement from an urn containing «Size» balls of which «PosEvents» are red.
+== Functions ==
+=== <div id="HyperGeometric''>HyperGeometric(trials, posEvents, size)</div> ===
+Use this to describe a variable whose outcome has a hyperGeometric distribution.
+=== <div id="ProbHyperGeometric">Prob{{Release||5.1|_}}HyperGeometric(k, trials, posEvents, size)</div> ===
+{{Release||5.1|To use, add the [[Distribution Densities Library]] to your model.}}
+Returns the probability of outcome «k». It is given by
+:<math>p(k) = { {\binom{posEvents}{k} \binom{size-posEvents}{trials-k} } \over \binom{size}{trials} }</math>
+Suppose 20 out of 30 balls in a basket are red, and you draw 10 balls at random without replacement.  Then the probability of drawing exactly 7 red balls is given by:
+:<code>ProbHyperGeometric(7, 10, 20, 30) &rarr; 0.3096</code>
+=== <div id="CumHyperGeometric">CumHyperGeometric(k, trials, posEvents, size)</div> ===
+{{Release||5.1|To use, add the [[Distribution Densities Library]] to your model.}}
+The cumulative probability function for the hyperGeometric distribution. Its value is equal to
+:<math>F(k) = \sum_{i=0}^{k} { {\binom{posEvents}{i} \binom{size-posEvents}{trials-i} } \over \binom{size}{trials} }</math>
+Use this function when computing the p-Value for a hyperGeometric statistical test.
+=== <div id="CumHyperGeometricInv">CumHyperGeometricInv(p, trials, posEvents, size)</div> ===
+{{Release||5.1|To use, add the [[Distribution Densities Library]] to your model.}}
+The inverse cumulative probability function for the hyperGeometric distribution
-Thus, the parameters are:
+=== Parameters ===
-;«Trials»: The sample size -— e.g., the number of balls drawn from an urn without replacement. Cannot be larger than «Size».
+;«trials»: The sample size -— e.g., the number of balls drawn from an urn without replacement. Cannot be larger than «Size».
-;«PosEvents»: The total number of successful events in the population -- e.g, the number of red balls in the urn.
+;«posEvents»: The total number of successful events in the population -- e.g, the number of red balls in the urn.
-;«Size»: The population size -- e.g., the total number of balls in the urn, red and non-red.
+;«size»: The population size -- e.g., the total number of balls in the urn, red and non-red.
-== Library ==
+=== History ===
-Distributions
+* The analytic functions (ProbHyperGeometric, CumHyperGeometric, and CumHyperGeometricInv) were added as built-in functions in [[Analytica 5.2]].
+* In [[Analytica 5.1]] or earlier, the analytic functions require you to add the [[Distributions Density Library]] to your model.
 == See Also ==
-* [[Geometric]]
+* [[Geometric distribution]]{{Release||5.1|
 * [[Prob_HyperGeometric]]
 * [[Prob_Geometric]]
 * [[CumGeometric]]
-* [[CumGeometricInv]]
+* [[CumGeometricInv]]}}
 * [[Parametric discrete distributions]]
 * [[Distribution Densities Library]]