formulasearchengine - User contributions [en]

Log-polar coordinates

2014-09-12T17:52:17Z

172.251.103.105: /* Discrete geometry */

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ty+item+shouldn%27t+scare+you.+Single+snap+retrieval+on+the+Cold+Steel+Kukri%0D%0A%0D%0Avery+best+machete+for+survival+very+best+survival+machete+survival+machete+Survival+Knife+All+The+Ideal+To+Pick+The+Very+best+Survival+Machete+greatest+machete+Machete+The+Machete+Picking+a+%5Bhttp%3A%2F%2Fphotobucket.com%2Fimages%2FSurvival+Survival%5D+Machete+Ideal+Survival+Machete+Models+Prime+modest+survival+machetes+Survival+Style+Bolo+Machete+Latin+Style+Survival+Machete+Click+Right+here+for+the+Top+Machetes+on+Amazon+How+to+Choose+the+Greatest+Survival+Machete+Gator+Machete+Machete+Blade+Gerber+Gator+Machete+Junior+and+Ka-Bar+Black+Kukri+Machete+Greatest+Obtain+Ontario+Knife+8335+Machete+Black+for+Unique+Now+%E2%80%93+Survival+Knife+Gator+Machete+Pro+The+Machete+Blade+The+Pocket+Survival+Guide+Survival+Gear+Wholesale+Survival+Machete+What+is+the+finest+survival+machete%3F+This+is+the+ultimate+guide+to+selecting+the+very+best+survival+machete.+For+a+list+of+the+ideal+machetes%2C+take+a+look+at+-survival-machete%2F+Survival+Books+Gerber%0D%0A%0D%0AWorrying+a+lot+about+the+probabilities+of+illness+they+set+about+obtaining+natural+remedies.+When+they+killed+animals%2C+they+would+reduce+open+its+stomach+collected+the+gastric+juice+and+dried+this+in+the+sun.+They+ground+the+solidified+mass+into+a+greenish-white+powder+%26+kept+it+in+a+bottle%2C+it+proved+an+exceptional+gastric+tonic.+Magnifying+lens+for+beginning+fire+%28The+magnesium+or+steel+wool+is+a+backup+to+the+lens+-+the+lens+does+not+work+when+it+really+is+overcast+or+dark+and+it+is+not+valuable+if+all+the+things+is+soaking+wet%2C+but+you+can+run+out+of+magnesium+powder+and+the+bars+are+murder+on+a+knife.+The+lens+in+no+way+runs+out.%29+In+the+west%2C+we+train%2C+we+educate+our+young+guys+to+the+very+best+of+our+potential.+And+in+the+finish+the+practical+experience+of+combat+is+their+finishing+college.&create=Create 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Isaac Roberts

2014-01-29T23:49:59Z

172.251.194.102: /* Interest in astronomy */

In [[population genetics]], '''Ewens' sampling formula''', describes the probabilities associated with counts of how many different [[allele]]s are observed a given number of times in the sample.

==Definition==
Ewens' sampling formula, introduced by [[Warren Ewens]], states that under certain conditions (specified below), if a random sample of ''n'' [[gamete]]s is taken from a population and classified according to the [[gene]] at a particular [[locus (genetics)|locus]] then the [[probability]] that there are ''a''1 [[allele]]s represented once in the sample, and ''a''2 alleles represented twice, and so on, is

:<math>\operatorname{Pr}(a_1,\dots,a_n; \theta)={n! \over \theta(\theta+1)\cdots(\theta+n-1)}\prod_{j=1}^n{\theta^{a_j} \over j^{a_j} a_j!},</math>

for some positive number ''θ'', whenever ''a''1, ..., ''a''''n'' is a sequence of nonnegative integers such that

:<math>a_1+2a_2+3a_3+\cdots+na_n=n.\,</math>

The phrase "under certain conditions", used above, must of course be made precise. The assumptions are:
* The sample size ''n'' is small by comparison to the size of the whole population; and
* The population is in statistical equilibrium under [[mutation]] and [[genetic drift]] and the role of selection at the locus in question is negligible; and
* Every mutant allele is novel. (See also [[idealised population]].)

This is a [[probability distribution]] on the set of all [[integer partition|partitions of the integer]] ''n''. Among probabilists and statisticians it is often called the '''multivariate Ewens distribution'''.

== Mathematical properties ==

When ''θ'' = 0, the probability is 1 that all ''n'' genes are the same. When ''θ'' = 1, then the distribution is precisely that of the integer partition induced by a uniformly distributed [[random permutation]]. As ''θ'' → ∞, the probability that no two of the ''n'' genes are the same approaches 1.

This family of probability distributions enjoys the property that if after the sample of ''n'' is taken, ''m'' of the ''n'' gametes are chosen without replacement, then the resulting probability distribution on the set of all partitions of the smaller integer ''m'' is just what the formula above would give if ''m'' were put in place of ''n''.

The Ewens distribution arises naturally from the [[Chinese restaurant process]].

==See also==
* [[Coalescent theory]]
* [[Unified neutral theory of biodiversity]]

{{inline|date=August 2011}}
==Notes==
* Warren Ewens, "The sampling theory of selectively neutral alleles", ''Theoretical Population Biology'', volume 3, pages 87–112, 1972.
* J.F.C. Kingman, "Random partitions in population genetics", ''Proceedings of the Royal Society of London, Series B, Mathematical and Physical Sciences'', volume 361, number 1704, 1978.
* S. Tavare and W. J. Ewens, [http://www.cmb.usc.edu/people/stavare/STpapers-pdf/TE97.pdf "The Multivariate Ewens distribution"]. (1997, Chapter 41 from the reference below).
* N.L. Johnson, S. Kotz, and N. Balakrishnan (1997) ''Discrete Multivariate Distributions'', Wiley. ISBN 0-471-12844-9.

{{ProbDistributions|multivariate}}

{{DEFAULTSORT:Ewens's Sampling Formula}}
[[Category:Probability distributions]]
[[Category:Population genetics]]
[[Category:Discrete distributions]]

Calculator input methods

2013-12-20T08:10:18Z

172.251.178.32: /* Reverse Polish notation (RPN) (postfix notation) */

{{Probability fundamentals}}

In [[probability theory]], the '''complement''' of any [[event (probability theory)|event]] ''A'' is the event [not ''A''], i.e. the event that ''A'' does not occur.<ref>Robert R. Johnson, Patricia J. Kuby: ''Elementary Statistics''. Cengage Learning 2007, ISBN 978-0-495-38386-4, p. 229 ({{Google books|DWCAh7jWO98C|restricted online copy|page=229}})</ref> The event ''A'' and its complement [not ''A''] are [[mutually exclusive]] and [[Collectively exhaustive events|exhaustive]]. Generally, there is only one event ''B'' such that ''A'' and ''B'' are both mutually exclusive and exhaustive; that event is the complement of ''A''. The complement of an event ''A'' is usually denoted as ''A′'', ''Ac'' or ''{{overline|A}}''.

For example, if a typical coin is tossed and one assumes that it cannot land on its edge, then it can either land showing "heads" or "tails." Because these two [[Outcome (probability)|outcomes]] are mutually exclusive (i.e. the coin cannot simultaneously show both heads and tails) and collectively exhaustive (i.e. there are no other possible outcomes not represented between these two), they are therefore each other's complements. This means that [heads] is logically equivalent to [not tails], and [tails] is equivalent to [not heads].

==Complement rule==
In a [[Experiment (probability theory)|random experiment]], the probabilities of all possible events (the [[sample space]]) must total to 1— that is, some outcome must occur on every trial. For two events to be complements, they must be [[Collectively exhaustive events|collectively exhaustive]], together filling the entire sample space. Therefore, the probability of an event's complement must be [[Unity (mathematics)|unity]] minus the
probability of the event.<ref name="yates">{{cite book | last = Yates | first = Daniel S. | last2 = Moore | first2 = David S | last3 = Starnes | first3 = Daren S. | year = 2003 | title = The Practice of Statistics | edition = 2nd | publisher = [[W. H. Freeman and Company|Freeman]] | location = New York | url = http://bcs.whfreeman.com/yates2e/ | isbn = 978-0-7167-4773-4}}</ref>{{rp|342-343}} That is, for an event ''A'',
:<math>P(A^c) = 1 - P(A).</math>
Equivalently, the probabilities of an event and its complement must always total to 1. This does not, however, mean that ''any'' two events whose probabilities total to 1 are each other's complements; complementary events must also fulfill the condition of [[Mutually exclusive events|mutual exclusivity]].

===Example of the utility of this concept===
Suppose one throws an ordinary six-sided die eight times. What is the probability that one sees a "1" at least once?

It may be tempting to say that

: Pr(["1" on 1st trial] or ["1" on second trial] or ... or ["1" on 8th trial])

:= Pr("1" on 1st trial) + Pr("1" on second trial) + ... + P("1" on 8th trial)

:= 1/6 + 1/6 + ... + 1/6.

:= 8/6 = 1.3333... (...and this is clearly wrong.)

That cannot be right because a probability cannot be more than 1. The technique is wrong because the eight events whose probabilities got added are not mutually exclusive.

Instead one may find the probability of the complementary event and subtract it from 1, thus:

: Pr(at least one "1") = 1 − Pr(no "1"s)

:= 1 − Pr([no "1" on 1st trial] and [no "1" on 2nd trial] and ... and [no "1" on 8th trial])

:= 1 − Pr(no "1" on 1st trial) × Pr(no "1" on 2nd trial) × ... × Pr(no "1" on 8th trial)

:= 1 −(5/6) × (5/6) × ... × (5/6)

:= 1 − (5/6)8

:= 0.7674...

==See also==
*[[Negation|Logical complement]]
*[[Exclusive disjunction]]
*[[Binomial probability]]

==References==
{{reflist}}

==External links==
*[http://highered.mcgraw-hill.com/sites/dl/free/0072549076/79746/ch04_p175.pdf ''Complementary events''] - (free) page from probability book of [[McGraw-Hill]]

{{DEFAULTSORT:Complementary Event}}
[[Category:Probability theory]]