Dirichlet-multinomial distribution - Revision history

71.65.245.240: /* For a set of individual outcomes */ Fixed incorrect formula.

2014-11-15T22:47:28Z

For a set of individual outcomes: Fixed incorrect formula.

← Older revision		Revision as of 00:47, 16 November 2014
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130.113.148.220: /* Joint distribution */

2014-02-20T20:56:38Z

Joint distribution

Show changes

107.0.83.66: insert missing w_{dn} from an example/* Multiple Dirichlet priors with the same hyperprior, with dependent children */

2014-01-15T23:21:40Z

insert missing w_{dn} from an exampleMultiple Dirichlet priors with the same hyperprior, with dependent children

← Older revision		Revision as of 01:21, 16 January 2014
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	~~Emilia Shryock~~ is my title ~~but you~~ can ~~call me something you like~~. ~~He utilized to~~ be ~~unemployed but now he~~ is a ~~meter reader~~. ~~For~~ a ~~whilst I've been~~ in ~~South Dakota~~ and ~~my parents reside close~~ by. ~~To collect cash~~ is ~~what her family members and her enjoy~~.<br><br>~~my web-site .~~.. [~~http~~://~~dns125~~.~~dolzer~~.at/?q=~~node~~/~~696 at home std test~~]		In [[mathematics]], the '''complex conjugate root theorem''' states that if ''P'' is a [[polynomial]] in one variable with [[Real numbers\|real]] [[coefficients]], and ''a'' + ''bi'' is a [[root of a function\|root]] of ''P'' with ''a'' and ''b'' real numbers, then its [[complex conjugate]] ''a'' − ''bi'' is also a root of ''P''.<ref>{{cite book\|title=Maynooth Mathematical Olympiad Manual\|author=Anthony G. O'Farell and Gary McGuire\|pages=104\|chapter=Complex numbers, 8.4.2 Complex roots of real polynomials\|year=2002\|publisher=Logic Press \|isbn=0954426908}} Preview available at [http://books.google.com/books?q=Maynooth+Mathematical+Olympiad+Manual&ots=xQ0hpAQkpc&sa=X&oi=print&ct=title Google books]</ref>

			It follows from this (and the [[fundamental theorem of algebra]]), that if the degree of a real polynomial is odd, it must have at least one real root.<ref name=Jeffrey>{{cite book\|title=Complex Analysis and Applications\|author=Alan Jeffrey\|chapter=Analytic Functions\|pages=22–23\|year=2005\|publisher=CRC Press\|id=ISBN 158488553X}}</ref> That fact can also be proven by using the [[intermediate value theorem]].

			== Examples and consequences ==
			* The polynomial ''x''<sup>2</sup> + 1 = 0 has roots ±''i''.
			* Any real square [[matrix(mathematics)\|matrix]] of odd degree has at least one real [[eigenvalue]]. For example, if the matrix is [[orthogonal matrix\| orthogonal]], then 1 or −1 is an eigenvalue.
			* The polynomial
			::<math> x^3 - 7x^2 + 41x - 87\, </math>
			:has roots
			::<math> 3,\, 2 + 5i,\, 2 - 5i,</math>
			:and thus can be factored as
			::<math> (x - 3)(x - 2 - 5i)(x - 2 + 5i).\,</math>
			:In computing the product of the last two factors, the imaginary parts cancel, and we get
			::<math> (x - 3)(x^2 - 4x + 29).\,</math>
			:The non-real factors come in pairs which when multiplied give quadratic polynomials with real coefficients. Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the [[fundamental theorem of algebra]]), it follows that every polynomial with real coefficients can be factored into factors of degree no higher than 2: just 1st-degree and quadratic factors.

			=== Corollary on odd-degree polynomials ===
			It follows from the present theorem and the [[fundamental theorem of algebra]] that if the degree of a real polynomial is odd, it must have at least one real root.<ref name="Jeffrey"/>

			This can be proved as follows.
			*Since non-real complex roots come in conjugate pairs, there are an even number of them;
			*But a polynomial of odd degree has an odd number of roots;
			*Therefore some of them must be real.

			This requires some care in the presence of [[multiple root]]s; but a complex root and its conjugate do have the same [[Multiplicity (mathematics)\|multiplicity]] (and this [[lemma (mathematics)\|lemma]] is not hard to prove). It can also be worked around by considering only [[irreducible polynomial]]s; any real polynomial of odd degree must have an irreducible factor of odd degree, which (having no multiple roots) must have a real root by the reasoning above.

			This corollary can also be proved directly by using the [[intermediate value theorem]].

			== Simple proof ==
			One proof of the theorem is as follows:<ref name=Jeffrey />

			Consider the polynomial

			: <math>P(z) = a_0 + a_1z + a_2z^2 + \cdots + a_nz^n</math>

			where all ''a''<sub>''r''</sub> are real. Suppose some complex number ''ζ'' is a root of ''P'', that is ''P''(''ζ'') = 0. It needs to be shown that
			: <math>P(\overline{\zeta}) = 0</math>

			as well.

			If ''P''(''ζ'') = 0, then

			: <math>a_0 + a_1\zeta + a_2\zeta^2 + \cdots + a_n\zeta^n = 0\,</math>

			which can be put as

			: <math>\sum_{r=0}^n a_r\zeta^r = 0.</math>

			Now
			: <math>P(\overline{\zeta}) = \sum_{r=0}^n a_r\left(\overline{\zeta}\right)^r</math>

			and given the [[Complex_conjugation#Properties\|properties of complex conjugation]],

			: <math>\sum_{r=0}^n a_r\left(\overline{\zeta}\right)^r = \sum_{r=0}^n a_r \overline{\zeta^r} = \sum_{r=0}^n \overline{a_r\zeta^r} = \overline {\sum_{r=0}^n a_r\zeta^r}.</math>

			Since,
			: <math>\overline {\sum_{r=0}^n a_r\zeta^r} = \overline{0}</math>

			it follows that

			: <math>\sum_{r=0}^n a_r\left(\overline{\zeta}\right)^r = \overline{0} = 0.</math>

			That is,

			: <math>P(\overline{\zeta}) = a_0 + a_1\overline{\zeta} + a_2\left(\overline{\zeta}\right)^2 + \cdots + a_n\left(\overline{\zeta}\right)^n = 0.</math>

			== Notes ==
			<div class="references-small"><references /></div>

			[[Category:Theorems in complex analysis]]
			[[Category:Polynomials]]
			[[Category:Theorems in algebra]]
			[[Category:Articles containing proofs]]

98.245.1.79: Correctionword to word.

2012-07-19T00:59:56Z

Correctionword to word.

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