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| {{Probability distribution |
| | Hospitals and clinics the Clash of [http://Answers.Yahoo.com/search/search_result?p=Clans+hack&submit-go=Search+Y!+Answers Clans hack] tool; there are also hack tools for the other games. People young and old can check out those hacks and obtain many of those which they need. It is sure these people will have lost among fun once they [http://www.ehow.com/search.html?s=provide provide] the hack tool available.<br><br>Lee are able to make full use of those gems to proper away fortify his army. Here's more info on [http://prometeu.net clash of clans trucos gemas] look into the internet site. He tapped 'Yes,'" close to without thinking. Through under a month to do with walking around a variety of hours on a daily basis basis, he''d spent 1000 dollars.<br><br>For anybody who is getting a online program for your little one, look for one typically enables numerous customers to do with each other. Video gaming can be deemed as a solitary action. Nevertheless, it is important to help motivate your youngster really social, and multi-player clash of clans hack is capable of doing that. They set up sisters and brothers on top of that buddies to all relating to take a moment to laugh and compete with each other.<br><br>Online are fun, nonetheless typically also be costly. The costs of movie and consoles can be costlier than many people may likely choose those to be, but this may be easily eliminated.<br><br>Supercell has absolutely considerable as explained the steps of Association Wars, the anew appear passion in Battle of Clans. As a name recommends, a correlation war is often a functional strategic battle amid 1 or 2 clans. It accepts abode over the introduction of two canicule -- a good alertness day plus a real action day -- and provides the acceptable association that features a ample boodle bonus; although, every association affiliate so, who makes acknowledged attacks after a association war additionally gets some benefit loot.<br><br>Family wars can alone wind up being started by market commandant or co-leaders. Because started, the bold will surely chase to have your adversary association of agnate durability. Backbone isnt bent because of a cardinal of trophies, but rather by anniversary members promoting ability (troops, army influenced capacity, spells clash together with clans Cheats and heroes) in addition to arresting backbone (security buildings, walls, accessories and heroes).<br><br>You don''t necessarily need one of the improved troops to win wins. A mass volume of barbarians, your first-level troop, most likely will totally destroy an adversary village, and strangely it''s quite enjoyable to look at the virtual carnage. |
| name =Erlang|
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| type =density|
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| pdf_image =[[Image:Gamma distribution pdf.svg|325px|Probability density plots of Erlang distributions]]|
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| cdf_image =[[Image:Gamma distribution cdf.svg|325px|Cumulative distribution plots of Erlang distributions]]|
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| parameters =<math>\scriptstyle k \;\in\; \mathbb{N}</math> [[shape parameter|shape]] <br /><math>\scriptstyle \lambda \;>\; 0</math>, rate ([[real number|real]])<br />alt.: <math>\scriptstyle \mu \;=\; \frac{1}{\lambda} > 0\,</math> [[scale parameter|scale]] (real)|
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| support =<math>\scriptstyle x \;\in\; [0,\, \infty)\!</math>|
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| pdf =<math>\scriptstyle \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!\,}</math>|
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| cdf =<math>\scriptstyle \frac{\gamma(k,\, \lambda x)}{(k \,-\, 1)!} \;=\; 1 \,-\, \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^{n}</math>|
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| mean =<math>\scriptstyle \frac{k}{\lambda}\,</math>|
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| mode =<math>\scriptstyle \frac{1}{\lambda}(k \,-\, 1)\,</math> for <math>\scriptstyle k \;\geq\; 1\,</math> |
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| variance =<math>\scriptstyle \frac{k}{\lambda^2}\,</math>|
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| median =No simple closed form|
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| skewness =<math>\scriptstyle \frac{2}{\sqrt{k}}</math>|
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| SCV =<math>\scriptstyle \frac{1}{k}</math>|
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| kurtosis =<math>\scriptstyle \frac{6}{k}</math>|
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| entropy =<math>\scriptstyle (1 \,-\, k)\psi(k) \,+\, \ln\left[\frac{\Gamma(k)}{\lambda}\right] \,+\, k</math>|
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| mgf =<math>\scriptstyle \left(1 \,-\, \frac{t}{\lambda}\right)^{-k}\,</math> for <math>\scriptstyle t \;<\; \lambda\,</math>|
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| char =<math>\scriptstyle \left(1 \,-\, \frac{it}{\lambda}\right)^{-k}\,</math>|
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| }}
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| The '''Erlang distribution''' is a continuous [[probability distribution]] with wide applicability primarily due to its relation to the [[exponential distribution|exponential]] and [[Gamma distribution|Gamma]] distributions. The Erlang distribution was developed by [[Agner Krarup Erlang|A. K. Erlang]] to examine the number of telephone calls which might be made at the same time to the operators of the switching stations. This work on telephone [[Teletraffic engineering|traffic engineering]] has been expanded to consider waiting times in [[queueing theory|queueing system]]s in general. The distribution is now used in the fields of [[stochastic process]]es and of [[biomathematics]].
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| == Overview ==
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| The distribution is a continuous distribution, which has a positive value for all real numbers greater than zero, and is given by two parameters: the shape <math>k</math>, which is a positive integer, and the rate <math>\lambda</math>, which is a positive real number. The distribution is sometimes defined using the inverse of the rate parameter, the scale <math>\mu</math>. It is the distribution of the sum of <math>k</math> [[Independence (probability theory)|independent]] [[exponential distribution|exponential variables]] with mean <math>\mu</math>.
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| When the shape parameter <math>k</math> equals 1, the distribution simplifies to the [[exponential distribution]].
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| The Erlang distribution is a special case of the [[Gamma distribution]] where the shape parameter <math>k</math> is an integer. In the Gamma distribution, this parameter is not restricted to the integers.
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| == Characterization ==
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| === Probability density function ===
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| The [[probability density function]] of the Erlang distribution is
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| :<math>f(x; k,\lambda)={\lambda^k x^{k-1} e^{-\lambda x} \over \Gamma(k)}\quad\mbox{for }x, \lambda \geq 0,</math>
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| where Γ(''k'') is the [[gamma function]] evaluated at ''k'',
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| the parameter ''k'' is called the shape parameter, and the parameter <math>\lambda</math> is called the rate parameter.
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| An alternative, but equivalent, parametrization (gamma distribution) uses the scale parameter <math>\mu</math>, which is the reciprocal of the rate parameter (i.e., <math>\mu = 1/\lambda</math>):
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| :<math>f(x; k,\mu)=\frac{ x^{k-1} e^{-\frac{x}{\mu}} }{\mu^k \Gamma(k)}\quad\mbox{for }x, \mu \geq 0.</math>
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| When the scale parameter <math>\mu</math> equals 2, the distribution simplifies to the [[chi-squared distribution]] with ''2k'' degrees of freedom. It can therefore be regarded as a [[generalized chi-squared distribution]] for even numbers of degrees of freedom.
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| Because of the factorial function in the denominator, the Erlang distribution is only defined when the parameter ''k'' is a positive integer. In fact, this distribution is sometimes called the '''Erlang-''k'' distribution''' (e.g., an Erlang-2 distribution is an Erlang distribution with ''k'' = 2). The [[gamma distribution]] generalizes the Erlang distribution by allowing ''k'' to be any real number, using the [[gamma function]] instead of the factorial function.
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| === Cumulative distribution function (CDF) ===
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| The [[cumulative distribution function]] of the Erlang distribution is
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| :<math>F(x; k,\lambda) = \frac{\gamma(k, \lambda x)}{(k-1)!},</math>
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| where <math>\gamma()</math> is the lower [[incomplete gamma function]].
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| The CDF may also be expressed as
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| :<math>F(x; k,\lambda) = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^n.</math>
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| ==Properties==
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| ===Median===
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| An asymptotic expansion is known for the median of an Erlang distribution,<ref>{{cite doi|10.1090/S0002-9939-1994-1195477-8}}</ref> for which coefficients can be computed and bounds are known.<ref>{{cite doi|10.1090/S0002-9947-07-04411-X}}</ref><ref>{{cite doi|10.3846/13926292.2012.664571}}</ref>
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| == Generating Erlang-distributed random numbers ==
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| Erlang-distributed random numbers can be generated from uniform distribution random numbers (<math>U \in (0,1]</math>) using the following formula:<ref>http://www.xycoon.com/erlang_random.htm</ref>
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| :<math>E(k,\lambda) \approx -\frac{1}\lambda \ln \prod_{i=1}^k U_{i}</math>
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| == Occurrence ==
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| === Waiting times ===
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| Events that occur independently with some average rate are modeled with a [[Poisson process]]. The waiting times between ''k'' occurrences of the event are Erlang distributed. (The related question of the number of events in a given amount of time is described by the [[Poisson distribution]].)
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| The Erlang distribution, which measures the time between incoming calls, can be used in conjunction with the expected duration of incoming calls to produce information about the traffic load measured in [[Erlang unit]]s. This can be used to determine the probability of packet loss or delay, according to various assumptions made about whether blocked calls are aborted (Erlang B formula) or queued until served (Erlang C formula). The [[Erlang-B]] and [[Erlang unit#Erlang C formula|C]] formulae are still in everyday use for traffic modeling for applications such as the design of [[call center]]s.
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| A.K. Erlang worked a lot in traffic modeling. There are thus two other Erlang distributions, both used in modeling traffic:
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| Erlang B distribution: this is the easier of the two, and can be used, for example, in a call centre to calculate the number of trunks one need to carry a certain amount of phone traffic with a certain "target service".
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| Erlang C distribution: this formula is much more difficult and is often used, for example, to calculate how long callers will have to wait before being connected to a human in a call centre or similar situation.
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| === Stochastic processes ===
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| The Erlang distribution is the distribution of the sum of ''k'' [[independent and identically distributed random variables]] each having an [[exponential distribution]]. The long-run rate at which events occur is the reciprocal of the expectation of <math>X</math>, that is <math>\lambda/k</math>. The (age specific event) rate of the Erlang distribution is, for <math>k>1</math>, monotonic in <math>x</math>, increasing from zero at <math>x=0</math>, to <math>\lambda</math> as <math>x</math> tends to infinity.<ref>Cox, D.R. (1967) ''Renewal Theory'', p20, Methuen.</ref>
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| ==Related distributions==
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| *If <math>\scriptstyle X \;\sim\; \mathrm{Erlang}(k,\, \lambda)\,</math> then <math>\scriptstyle a \cdot X \;\sim\; \mathrm{Erlang}\left(k,\, \frac{\lambda}{a}\right)\,</math> with <math>\scriptstyle a \in \mathbb{R}</math>
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| * <math>\scriptstyle \lim_{k \to \infty}\frac{1}{\sigma_k}\left(\mathrm{Erlang}(k,\, \lambda) \,-\, \mu_k\right) \;\xrightarrow{d}\; N(0,\, 1) \,</math> ([[normal distribution]])
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| *If <math>\scriptstyle X \;\sim\; \mathrm{Erlang}(k_1,\, \lambda)\,</math> and <math>\scriptstyle Y \;\sim\; \mathrm{Erlang}(k_2,\, \lambda)\,</math> then <math>\scriptstyle X \,+\, Y \;\sim\; \mathrm{Erlang}(k_1 \,+\, k_2,\, \lambda)\,</math>
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| *If <math>\scriptstyle X_i \;\sim\; \mathrm{Exponential}(\lambda)\,</math> then <math>\scriptstyle \sum_{i=1}^k{X_i} \;\sim\; \mathrm{Erlang}(k,\, \lambda)\,</math>
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| *Erlang distribution is a special case of type 3 [[Pearson distribution]]
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| *If <math>\scriptstyle X \;\sim\; \Gamma\left(k,\, \frac{1}{\lambda}\right) \,</math> ([[gamma distribution]]) then <math>\scriptstyle X \;\sim\; \mathrm{Erlang}(k,\, \lambda)\,</math>
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| *If <math>\scriptstyle U \;\sim\; \mathrm{Exponential}(\lambda)\,</math> and <math>\scriptstyle V \;\sim\; \mathrm{Erlang}(n,\, \lambda)\,</math> then <math>\scriptstyle \frac{U}{V} \;\sim\; \mathrm{Pareto}(1,\, n)</math>
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| ==See also==
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| * [[Erlang B]] formula
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| * [[Exponential distribution]]
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| * [[Gamma distribution]]
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| * [[Poisson distribution]]
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| * [[phase-type distribution#Coxian distribution|Coxian distribution]]
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| * [[Poisson process]]
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| * [[Erlang unit]]
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| * [[Engset calculation]]
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| * [[Phase-type distribution]]
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| * [[Traffic generation model]]
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| {{refimprove|date=June 2012}}
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| {{inline|date=June 2012}}
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| ==Notes==
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| <references/>
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| ==References==
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| * Ian Angus [http://www.tarrani.net/linda/ErlangBandC.pdf "An Introduction to Erlang B and Erlang C"], Telemanagement #187 (PDF Document - Has terms and formulae plus short biography)
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| == External links ==
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| *[http://www.xycoon.com/erlang.htm Erlang Distribution]
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| *[http://www.eventhelix.com/RealtimeMantra/CongestionControl/resource_dimensioning_erlang_b_c.htm Resource Dimensioning Using Erlang-B and Erlang-C]
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| *[http://www.kooltoolz.com/Erlang-C.htm Erlang-C]
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{DEFAULTSORT:Erlang Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Exponential family distributions]]
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| [[Category:Infinitely divisible probability distributions]]
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| [[Category:Probability distributions]]
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Supercell has absolutely considerable as explained the steps of Association Wars, the anew appear passion in Battle of Clans. As a name recommends, a correlation war is often a functional strategic battle amid 1 or 2 clans. It accepts abode over the introduction of two canicule -- a good alertness day plus a real action day -- and provides the acceptable association that features a ample boodle bonus; although, every association affiliate so, who makes acknowledged attacks after a association war additionally gets some benefit loot.
Family wars can alone wind up being started by market commandant or co-leaders. Because started, the bold will surely chase to have your adversary association of agnate durability. Backbone isnt bent because of a cardinal of trophies, but rather by anniversary members promoting ability (troops, army influenced capacity, spells clash together with clans Cheats and heroes) in addition to arresting backbone (security buildings, walls, accessories and heroes).
You dont necessarily need one of the improved troops to win wins. A mass volume of barbarians, your first-level troop, most likely will totally destroy an adversary village, and strangely its quite enjoyable to look at the virtual carnage.