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Undid revision 614519301 by 208.50.124.65 (talk) The Lemone hexagon has one non-complex solution
 
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{{Unreferenced|date=December 2009}}
== there are lotus. ==
A '''compound Poisson process''' is a continuous-time (random) [[stochastic process]] with jumps. The jumps arrive randomly according to a [[Poisson process]] and the size of the jumps is also random, with a specified probability distribution. A compound Poisson process, parameterised by a rate <math>\lambda > 0</math> and jump size distribution ''G'', is a process <math>\{\,Y(t) : t \geq 0 \,\}</math> given by


:<math>Y(t) = \sum_{i=1}^{N(t)} D_i</math>
In a lot of colorful lotus land records.<br><br>ordinary barrel. Also half-human Heights barrel. Half a bucket can be.<br><br>and this pond. Nearly circular shape. About nearly two meters in diameter, so big pond. Inside the one yuan heavy water. Estimated to be sufficient to store ten barrels of heavy water to one yuan.<br><br>both ends of [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン 靴 メンズ] the pond in the [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン バッグ] yuan heavy water. Zhang Lin just two colorful, colorful lotus really look the same as [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン セール] described in the reel. Lotus is divided into 288, lotus minimal, there are lotus.<br><br>'Huh?' Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン 靴] Yu suddenly surprised again.<br><br>'central pond is arranged a little to hide the magic circle.' Qin Yu force to explore space discovered. This pond central in the end what is being used to hide the magic array block.<br><br>the passage of time, then quickly passed more than three years, more than three years time. [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン バッグ] Liu Yuxi talks nine were sitting cross-legged cry in silence, Qin Yu himself sitting cross-legged in the room heard in silence.<br><br>retreat at the child in the Qin Yu. Suddenly sensed glare to light, when the struggle to open the eyes and yang
相关的主题文章：
<ul>
 
  <li>[http://www.effigies.com/cgi-bin/guestbook/guestbook.cgi http://www.effigies.com/cgi-bin/guestbook/guestbook.cgi]</li>
 
  <li>[http://www.technowomb.com/cgi-bin/guestbook/guestbook.cgi http://www.technowomb.com/cgi-bin/guestbook/guestbook.cgi]</li>
 
  <li>[http://www.fuxiaoliming.com/forum.php?mod=viewthread&tid=208121 http://www.fuxiaoliming.com/forum.php?mod=viewthread&tid=208121]</li>
 
</ul>


where, <math> \{\,N(t) : t \geq 0\,\}</math> is a [[Poisson process]] with rate <math>\lambda</math>, and <math> \{\,D_i : i \geq 1\,\}</math> are independent and identically distributed random variables, with distribution function ''G'', which are also independent of <math> \{\,N(t) : t \geq 0\,\}.\,</math>
== leading to a lot of other galaxies Chuansong Zhen. ==


When <math> D_i </math> are non-negative integer-valued random variable, then this compound Poisson process is named stuttering Poisson process which has the feature that two or more events occur in a very short time .
Seeking heart.<br><br>'Yu Huang Majesty broad-minded, will not ......'<br><br>see Emperor Xian Yu also persuade the Vatican, Qin Yu busy hand nonchalantly said: 'Well, I do not have time and クリスチャンルブタン 店舗 you nonsense, leave' Qin Yu finished the whole person speed channeling out, after a great diversion again disappeared.<br><br>Qin Yu next destination 'blue star.'<br><br>blue star, as Angelica galactic core planet, this planet in Star Chuansong Zhen course very much, traffic developed, leading to a lot of other galaxies Chuansong Zhen.<br>'Friends dark astral ......'<br>Emperor Xian Yu Vatican wants to follow to catch up.<br><br>'seems only to let クリスチャンルブタン 偽物 Yu Huang personally resolved, I do not know where now arrived.' Bai star in his own to start a summons to Yu Huang, unfortunately there is no communications success.<br><br>Yu クリスチャンルブタン メンズ 靴 Wong and because he is too far away, only to be summoned クリスチャンルブタン 偽物 within the 'maze Communications' success. Vatican last Emperor Xian Yu Lian cents or so ice will matter to Yu Huang Di reporting.<br><br>Emperor Xian Lian from クリスチャンルブタン通販 the ice to his arraignment
相关的主题文章：
<ul>
 
  <li>?aid=105609</li>
 
  <li>?aid=2</li>
 
  <li>?aid=393439</li>
 
</ul>


==Properties of the compound Poisson process==
== palace courtyard alone among the Qin Yu ==
Using [[law of total expectation|conditional expectation]], the [[expected value]] of a compound Poisson process can be calculated using a result known as [[Wald's equation]] as:


:<math>\,E(Y(t)) = E(E(Y(t)|N(t))) = E(N(t)E(D)) = E(N(t))E(D) = \lambda t E(D).</math>
Had lost a war and key central repair, while wind figs, QIN talk about the problem of comprehension aspects. As for the injury itself, Qin Yu, has long been [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン 店舗 東京] a popular tears repaired.<br><br>palace courtyard alone among the Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_10.php クリスチャンルブタン 値段] Yu, Yu Qin, Qin Feng, Qin Zheng, QIN, wind figs five people sitting around the stone table, at the moment on the stone table stood three Cheats, it had left Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン セール] Yu three comprehension Cheats.<br><br>'Kohane, these three cheats you, you and I have seen the wind uncle, you wind uncle say, this three Cheats comprehension of each of the circles are uppermost comprehension Cheats, you wind uncle are now discarded Over the past practice of law, the use of this method of the above first comprehension. 'QIN smile.<br><br>wind figs also exclaimed: 'Kohane is the original Munekado secret pass to [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン 東京] practice law, but I also know the secret of Yuan Ying gas only, but obviously not as good as these cheats so deep so proud of mysterious practice of law, it is estimated. Munekado town than we have severe cases of canon. '<br><br>Qin Yu [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_0.php クリスチャンルブタン 店舗] is not talking, but with one hand stretched, spiritual goods is five knives and a black rifles floating
 
相关的主题文章：
Making similar use of the [[law of total variance]], the [[variance]] can be calculated as:
<ul>
:<math>
 
\begin{align}
  <li>[http://www.crowningtech.com/plus/feedback.php?aid=251 http://www.crowningtech.com/plus/feedback.php?aid=251]</li>
\operatorname{var}(Y(t)) &= E(\operatorname{var}(Y(t)|N(t))) + \operatorname{var}(E(Y(t)|N(t))) \\
 
&= E(N(t)\operatorname{var}(D)) + \operatorname{var}(N(t)E(D)) \\
  <li>[http://hhtea.com/plus/feedback.php?aid=1321 http://hhtea.com/plus/feedback.php?aid=1321]</li>
&= \operatorname{var}(D)E(N(t)) + E(D)^2 \operatorname{var}(N(t)) \\
 
&= \operatorname{var}(D)\lambda t + E(D)^2\lambda t \\
  <li>[http://www2s.biglobe.ne.jp/~ton/cgbbs/joyful.cgi http://www2s.biglobe.ne.jp/~ton/cgbbs/joyful.cgi]</li>
&= \lambda t(\operatorname{var}(D) + E(D)^2) \\
 
&= \lambda t E(D^2).
</ul>
\end{align}
</math>
 
Lastly, using the [[law of total probability]], the [[moment generating function]] can be given as follows:
:<math>\,\Pr(Y(t)=i) = \sum_{n} \Pr(Y(t)=i|N(t)=n)\Pr(N(t)=n) </math>
 
:<math>
\begin{align}
E(e^{sY}) & = \sum_i e^{si} \Pr(Y(t)=i) \\
& = \sum_i e^{si} \sum_{n} \Pr(Y(t)=i|N(t)=n)\Pr(N(t)=n) \\
& = \sum_n \Pr(N(t)=n) \sum_i e^{si} \Pr(Y(t)=i|N(t)=n) \\
& = \sum_n \Pr(N(t)=n) \sum_i e^{si}\Pr(D_1 + D_2 + \cdots + D_n=i) \\
& = \sum_n \Pr(N(t)=n) M_D(s)^n \\
& = \sum_n \Pr(N(t)=n) e^{n\ln(M_D(s))} \\
& = M_{N(t)}(\ln(M_D(s)) \\
& = e^{\lambda t \left ( M_D(s) - 1\right ) }.
\end{align}
</math>
 
==Exponentiation of measures==
Let ''N'', ''Y'', and ''D'' be as above. Let ''μ'' be the probability measure according to which ''D'' is distributed, i.e.
 
:<math>\mu(A) = \Pr(D \in A).\,</math>
 
Let ''δ''<sub>0</sub> be the trivial probability distribution putting all of the mass at zero. Then the [[probability distribution]] of ''Y''(''t'') is the measure
 
:<math>\exp(\lambda t(\mu - \delta_0))\,</math>
 
where the exponential exp(''ν'') of a finite measure ''ν'' on [[Borel set|Borel subsets]] of the [[real number|real line]] is defined by
 
:<math>\exp(\nu) = \sum_{n=0}^\infty {\nu^{*n} \over n!}</math>
 
and
 
:<math> \nu^{*n} = \underbrace{\nu * \cdots *\nu}_{n \text{ factors}}</math>
 
is a [[convolution]] of measures, and the series converges [[convergence of random variables|weakly]].
 
==Fitting a compound Poisson process==
 
The parameters for independent observations of a compound Poisson process can be chosen using a [[maximum likelihood estimator]] using Simar's algorithm,<ref>{{cite doi|10.1214/aos/1176343651}}</ref> which has been shown to converge.<ref>{{cite doi|10.1214/aos/1176345890}}</ref>
 
==See also==
* [[Poisson process]]
* [[Poisson distribution]]
* [[Non-homogeneous Poisson process]]
* [[Fractional Poisson process]]
* [[Campbell's formula]] for the [[moment generating function]] of a compound Poisson process
 
==References==
 
{{reflist}}
 
{{Stochastic processes}}
 
{{DEFAULTSORT:Compound Poisson Process}}
[[Category:Poisson processes]]
 
[[de:Poisson-Prozess#Zusammengesetzte_Poisson-Prozesse]]

Latest revision as of 18:30, 26 June 2014

there are lotus.

In a lot of colorful lotus land records.

ordinary barrel. Also half-human Heights barrel. Half a bucket can be.

and this pond. Nearly circular shape. About nearly two meters in diameter, so big pond. Inside the one yuan heavy water. Estimated to be sufficient to store ten barrels of heavy water to one yuan.

both ends of クリスチャンルブタン 靴 メンズ the pond in the クリスチャンルブタン バッグ yuan heavy water. Zhang Lin just two colorful, colorful lotus really look the same as クリスチャンルブタン セール described in the reel. Lotus is divided into 288, lotus minimal, there are lotus.

'Huh?' Qin クリスチャンルブタン 靴 Yu suddenly surprised again.

'central pond is arranged a little to hide the magic circle.' Qin Yu force to explore space discovered. This pond central in the end what is being used to hide the magic array block.

the passage of time, then quickly passed more than three years, more than three years time. クリスチャンルブタン バッグ Liu Yuxi talks nine were sitting cross-legged cry in silence, Qin Yu himself sitting cross-legged in the room heard in silence.

retreat at the child in the Qin Yu. Suddenly sensed glare to light, when the struggle to open the eyes and yang 相关的主题文章：

leading to a lot of other galaxies Chuansong Zhen.

Seeking heart.

'Yu Huang Majesty broad-minded, will not ......'

see Emperor Xian Yu also persuade the Vatican, Qin Yu busy hand nonchalantly said: 'Well, I do not have time and クリスチャンルブタン 店舗 you nonsense, leave' Qin Yu finished the whole person speed channeling out, after a great diversion again disappeared.

Qin Yu next destination 'blue star.'

blue star, as Angelica galactic core planet, this planet in Star Chuansong Zhen course very much, traffic developed, leading to a lot of other galaxies Chuansong Zhen.
'Friends dark astral ......'
Emperor Xian Yu Vatican wants to follow to catch up.

'seems only to let クリスチャンルブタン 偽物 Yu Huang personally resolved, I do not know where now arrived.' Bai star in his own to start a summons to Yu Huang, unfortunately there is no communications success.

Yu クリスチャンルブタン メンズ 靴 Wong and because he is too far away, only to be summoned クリスチャンルブタン 偽物 within the 'maze Communications' success. Vatican last Emperor Xian Yu Lian cents or so ice will matter to Yu Huang Di reporting.

Emperor Xian Lian from クリスチャンルブタン通販 the ice to his arraignment 相关的主题文章：

  • ?aid=105609
  • ?aid=2
  • ?aid=393439

palace courtyard alone among the Qin Yu

Had lost a war and key central repair, while wind figs, QIN talk about the problem of comprehension aspects. As for the injury itself, Qin Yu, has long been クリスチャンルブタン 店舗 東京 a popular tears repaired.

palace courtyard alone among the Qin クリスチャンルブタン 値段 Yu, Yu Qin, Qin Feng, Qin Zheng, QIN, wind figs five people sitting around the stone table, at the moment on the stone table stood three Cheats, it had left Qin クリスチャンルブタン セール Yu three comprehension Cheats.

'Kohane, these three cheats you, you and I have seen the wind uncle, you wind uncle say, this three Cheats comprehension of each of the circles are uppermost comprehension Cheats, you wind uncle are now discarded Over the past practice of law, the use of this method of the above first comprehension. 'QIN smile.

wind figs also exclaimed: 'Kohane is the original Munekado secret pass to クリスチャンルブタン 東京 practice law, but I also know the secret of Yuan Ying gas only, but obviously not as good as these cheats so deep so proud of mysterious practice of law, it is estimated. Munekado town than we have severe cases of canon. '

Qin Yu クリスチャンルブタン 店舗 is not talking, but with one hand stretched, spiritual goods is five knives and a black rifles floating 相关的主题文章：

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