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| In [[probability theory]], the '''probability-generating function''' of a [[discrete random variable]] is a [[power series]] representation (the [[generating function]]) of the [[probability mass function]] of the random variable. Probability-generating functions are often employed for their succinct description of the sequence of probabilities Pr(''X'' = ''i'') in the probability mass function for a [[random variable]] ''X'', and to make available the well-developed theory of power series with non-negative coefficients.
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| ==Definition==
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| === Univariate case ===
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| If ''X'' is a [[discrete random variable]] taking values in the non-negative [[integer]]s {0,1, ...}, then the ''probability-generating function'' of ''X'' is defined as
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| <ref>http://www.am.qub.ac.uk/users/g.gribakin/sor/Chap3.pdf</ref>
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| :<math>G(z) = \operatorname{E} (z^X) = \sum_{x=0}^{\infty}p(x)z^x,</math>
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| where ''p'' is the probability mass function of ''X''. Note that the subscripted notations ''G''<sub>''X''</sub> and ''p<sub>X</sub>'' are often used to emphasize that these pertain to a particular random variable ''X'', and to its distribution. The power series [[absolute convergence|converges absolutely]] at least for all [[complex number]]s ''z'' with |''z''| ≤ 1; in many examples the radius of convergence is larger.
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| === Multivariate case ===
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| If {{nowrap|''X'' {{=}} (''X''<sub>1</sub>,...,''X<sub>d</sub>'' )}} is a discrete random variable taking values in the ''d''-dimensional non-negative [[integer lattice]] {0,1, ...}<sup>''d''</sup>, then the ''probability-generating function'' of ''X'' is defined as
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| :<math>G(z) = G(z_1,\ldots,z_d)=\operatorname{E}\bigl (z_1^{X_1}\cdots z_d^{X_d}\bigr) = \sum_{x_1,\ldots,x_d=0}^{\infty}p(x_1,\ldots,x_d)z_1^{x_1}\cdots z_d^{x_d},</math>
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| where ''p'' is the probability mass function of ''X''. The power series converges absolutely at least for all complex vectors {{nowrap| ''z'' {{=}} (''z''<sub>1</sub>,...,''z<sub>d</sub>'' ) ∈ ℂ<sup>''d''</sup>}} with {{nowrap|max<nowiki>{|</nowiki>''z''<sub>1</sub><nowiki>|</nowiki>,...,<nowiki>|</nowiki>''z<sub>d</sub>'' <nowiki>|}</nowiki> ≤ 1}}.
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| ==Properties==
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| ===Power series===
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| Probability-generating functions obey all the rules of power series with non-negative coefficients. In particular, ''G''(1<sup>−</sup>) = 1, where ''G''(1<sup>−</sup>) = lim<sub>z→1</sub>''G''(''z'') [[One-sided limit|from below]], since the probabilities must sum to one. So the [[radius of convergence]] of any probability-generating function must be at least 1, by [[Abel's theorem]] for power series with non-negative coefficients.
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| ===Probabilities and expectations===
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| The following properties allow the derivation of various basic quantities related to ''X'':
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| 1. The probability mass function of ''X'' is recovered by taking [[derivative]]s of ''G''
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| :<math> p(k) = \operatorname{Pr}(X = k) = \frac{G^{(k)}(0)}{k!}.</math>
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| 2. It follows from Property 1 that if random variables ''X'' and ''Y'' have probability generating functions that are equal, ''G''<sub>''X''</sub> = ''G''<sub>''Y''</sub>, then ''p''<sub>''X''</sub> = ''p''<sub>''Y''</sub>. That is, if ''X'' and ''Y'' have identical probability-generating functions, then they have identical distributions.
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| 3. The normalization of the probability density function can be expressed in terms of the generating function by
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| :<math>\operatorname{E}(1)=G(1^-)=\sum_{i=0}^\infty f(i)=1.</math>
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| The [[expected value|expectation]] of ''X'' is given by
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| :<math> \operatorname{E}\left(X\right) = G'(1^-).</math> | |
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| More generally, the ''k''<sup>th</sup> [[factorial moment]], <math>\textrm{E}(X(X - 1) \cdots (X - k + 1))</math> of ''X'' is given by
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| :<math>\textrm{E}\left(\frac{X!}{(X-k)!}\right) = G^{(k)}(1^-), \quad k \geq 0.</math>
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| So the [[variance]] of ''X'' is given by
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| :<math>\operatorname{Var}(X)=G''(1^-) + G'(1^-) - \left [G'(1^-)\right ]^2.</math>
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| 4. <math>G_X(e^{t}) = M_X(t)</math> where ''X'' is a random variable, <math>G_X(t)</math> is the probability generating function (of ''X'') and <math>M_X(t)</math> is the [[moment-generating function]] (of ''X'') .
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| ===Functions of independent random variables===
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| Probability-generating functions are particularly useful for dealing with functions of [[statistical independence|independent]] random variables. For example:
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| * If ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>n</sub> is a sequence of independent (and not necessarily identically distributed) random variables, and
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| ::<math>S_n = \sum_{i=1}^n a_i X_i,</math>
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| :where the ''a''<sub>i</sub> are constants, then the probability-generating function is given by
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| ::<math>G_{S_n}(z) = \operatorname{E}(z^{S_n}) = \operatorname{E}(z^{\sum_{i=1}^n a_i X_i,}) = G_{X_1}(z^{a_1})G_{X_2}(z^{a_2})\cdots G_{X_n}(z^{a_n}).</math>
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| :For example, if
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| ::<math>S_n = \sum_{i=1}^n X_i,</math>
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| :then the probability-generating function, ''G''<sub>''Sn''</sub>(''z''), is given by
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| ::<math>G_{S_n}(z) = G_{X_1}(z)G_{X_2}(z)\cdots G_{X_n}(z).</math>
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| :It also follows that the probability-generating function of the difference of two independent random variables ''S'' = ''X''<sub>1</sub> − ''X''<sub>2</sub> is
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| ::<math>G_S(z) = G_{X_1}(z)G_{X_2}(1/z).</math>
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| *Suppose that ''N'' is also an independent, discrete random variable taking values on the non-negative integers, with probability-generating function ''G''<sub>''N''</sub>. If the ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>N</sub> are independent ''and'' identically distributed with common probability-generating function ''G''<sub>X</sub>, then
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| ::<math>G_{S_N}(z) = G_N(G_X(z)).</math>
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| :This can be seen, using the [[law of total expectation]], as follows:
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| ::<math> G_{S_N}(z) = \operatorname{E}(z^{S_N}) = \operatorname{E}(z^{\sum_{i=1}^N X_i}) = \operatorname{E}\big(\operatorname{E}(z^{\sum_{i=1}^N X_i}| N) \big) = \operatorname{E}\big( (G_X(z))^N\big) =G_N(G_X(z)).</math>
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| :This last fact is useful in the study of [[Galton–Watson process]]es.
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| *Suppose again that ''N'' is also an independent, discrete random variable taking values on the non-negative integers, with probability-generating function ''G''<sub>''N''</sub> and probability density <math>f_i = \Pr\{N = i\}</math>. If the ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>N</sub> are independent, but ''not'' identically distributed random variables, where <math>G_{X_i}</math> denotes the probability generating function of <math>X_i</math>, then
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| ::<math>G_{S_N}(z) = \sum_{i \ge 1} f_i \prod_{k=1}^i G_{X_i}(z).</math>
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| :For identically distributed ''X<sub>i</sub>'' this simplifies to the identity stated before. The general case is sometimes useful to obtain a decomposition of ''S<sub>N</sub>'' by means of generating functions.
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| ==Examples==
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| * The probability-generating function of a [[degenerate distribution|constant random variable]], i.e. one with Pr(''X'' = ''c'') = 1, is
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| ::<math>G(z) = \left(z^c\right). \, </math>
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| * The probability-generating function of a [[binomial distribution|binomial random variable]], the number of successes in ''n'' trials, with probability ''p'' of success in each trial, is
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| ::<math>G(z) = \left[(1-p) + pz\right]^n. \, </math>
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| :Note that this is the ''n''-fold product of the probability-generating function of a [[Bernoulli distribution|Bernoulli random variable]] with parameter ''p''.
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| * The probability-generating function of a [[negative binomial distribution|negative binomial random variable]] on {0,1,2 ...}, the number of failures until the ''r''th success with probability of success in each trial ''p'', is
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| ::<math>G(z) = \left(\frac{p}{1 - (1-p)z}\right)^r.</math>
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| :(Convergence for <math>|z| < \frac{1}{1-p}</math>).
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| :Note that this is the ''r''-fold product of the probability generating function of a [[geometric distribution|geometric random variable]] with parameter 1−''p'' on {0,1,2 ...}.
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| * The probability-generating function of a [[Poisson distribution|Poisson random variable]] with rate parameter λ is
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| ::<math>G(z) = \textrm{e}^{\lambda(z - 1)}.\;\,</math>
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| <!--
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| TO BE COMPLETED:
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| ==Joint probability-generating functions==
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| The concept of the probability-generating function for single random variables can be extended to the joint probability-generating function of two or more random variables.
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| Suppose that ''X'' and ''Y'' are both discrete random variables (not necessarily independent or identically distributed), again taking values on some subset of the non-negative integers. -->
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| ==Related concepts==
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| The probability-generating function is an example of a [[generating function]] of a sequence: see also [[formal power series]]. It is occasionally called the [[z-transform]] of the probability mass function.
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| Other generating functions of random variables include the [[moment-generating function]], the [[Characteristic function (probability theory)|characteristic function]] and the [[cumulant generating function]].
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| {{refimprove|date=April 2012}}
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| ==Notes==
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| {{reflist|refs=
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| }}
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| ==References==
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| *Johnson, N.L.; Kotz, S.; Kemp, A.W. (1993) ''Univariate Discrete distributions'' (2nd edition). Wiley. ISBN 0-471-54897-9 (Section 1.B9)
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| {{Theory of probability distributions}}
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| {{DEFAULTSORT:Probability-Generating Function}}
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| [[Category:Theory of probability distributions]]
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| [[Category:Generating functions]]
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Msvcr71.dll is an significant file which assists help Windows procedure different components of the system including significant files. Specifically, the file is chosen to help run corresponding files in the "Virtual C Runtime Library". These files are significant in accessing any settings that support the different applications and programs in the system. The msvcr71.dll file fulfills many significant functions; though it's not spared from getting damaged or corrupted. Once the file gets corrupted or damaged, the computer might have a difficult time processing and reading components of the system. However, consumers want not panic because this problem can be solved by following many procedures. And I will show we certain strategies about Msvcr71.dll.
Most of the reliable companies will offer a full income back guarantee. This signifies that we have the opportunity to get the money back if you find the registry cleaning has not delivered what we expected.
H/w related error handling - whenever hardware causes BSOD installing latest fixes for the hardware and/ or motherboard can assist. We could moreover add brand-new hardware that is compatible with the program.
If you feel we don't have enough cash at the time to upgrade, then the best way is to free up some room by deleting some of the unwelcome files plus folders.
The final step is to make sure that you clean the registry of your computer. The "registry" is a big database which shops significant files, settings & choices, and information. Windows reads the files it demands in purchase for it to run programs by this database. If the registry gets damaged, afflicted, or clogged up, then Windows will not be able to correctly access the files it needs for it to load up programs. As this happens, issues plus mistakes like the d3d9.dll error occur. To fix this and prevent future setbacks, you need to download and run a registry cleaning tool. The very recommended software is the "Frontline tuneup utilities".
Active X controls are used across the entire spectrum of computer plus web technologies. These controls are called the building blocks of the internet and as the glue which puts it all together. It is a standard that is used by all programmers to create the web more beneficial and interactive. Without these control standards there would basically be no public web.
Maybe you may be asking how come these windows XP error messages appear. Well, for you to be capable to understand the fix, you need to first recognize where those mistakes come from. There is this software called registry. A registry is software which shops everything on a PC from a general configuration, setting, information, and logs of escapades from installing to UN-installing, saving to deleting, plus a lot more alterations we do inside your program pass through it plus gets 'tagged' and saved as a easy file for healing reasons. Imagine it because a big recorder, a registrar, of all a records inside your PC.
So in summary, whenever comparing registry cleaning, make sure the one we choose offers you the following.A backup plus restore center, quickly procedure, automatic deletion facility, start-up administration, an convenient way of contact plus a funds back guarantee.