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| {{Probability distribution |
| | Do we want to lower the BMI (Body Mass Index)? Do you need to become slimmer? Do you need to have more lean muscle than fat? Follow these simple suggestions.<br><br>Keep in your mind, everyone's body is completely different, so finding how many calories the body requires on a daily basis may have several determining factors, nevertheless a primary one is how several occasions you exercise. You might employ an online Calorie Calculator to aid approximate the amount of calories a body must keep its existing fat. We can also consider struggling a online BMI index and see should you are underweight, regular, or obese for the body size.<br><br>He actually had sort 1.5 diabetes or Latent autoimmune diabetes of adults (LADA). Here is what Wikipedia says about LADA. Officially it can nevertheless be mentioned to be kind 1 diabetes. This really is a smaller post of mine, however you are able to observe the 2 videos below and you can also buy Dr. Cousen's book on curing diabetes.<br><br>The Body Mass Index is based on a person's fat inside proportion to their height plus age. The National Institute of Health provides a convenient online [http://safedietplansforwomen.com/bmi-calculator bmi calculator females] that automatically computes the value based found on the entered height plus weight. Results under 18.5 get into the "underweight" range, that may cause many wellness risks including malnurition and emaciation.<br><br>BMI is selected worldwide to determine if an individual is obese. Since BMI is chosen only for a screening tool, anybody whom says which you are at health risk before performing any different tests is lying. Doctors may screen individuals for wellness risks after seeing which their BMI is above normal.<br><br>There were 4 others that had kind 2 diabetes plus one with type 1 diabetes. After a month, the one with kind 1 diabetes was able to drop his insulin from 70 units right down to 5 units a day. With the 4 others, they were able to get off all of their medicine and have a normal blood glucose level meaning which they were cured.<br><br>In terms of its dodgy mathematics, the BMI is borderline junk research. However inside this regard, BMI is less egregious than Global Warming 'studies', inside that many of the big-name 'researchers' cherry-pick, hide, or even fabricate data. Click found on the link for my lengthy hub on the topic. |
| 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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Do we want to lower the BMI (Body Mass Index)? Do you need to become slimmer? Do you need to have more lean muscle than fat? Follow these simple suggestions.
Keep in your mind, everyone's body is completely different, so finding how many calories the body requires on a daily basis may have several determining factors, nevertheless a primary one is how several occasions you exercise. You might employ an online Calorie Calculator to aid approximate the amount of calories a body must keep its existing fat. We can also consider struggling a online BMI index and see should you are underweight, regular, or obese for the body size.
He actually had sort 1.5 diabetes or Latent autoimmune diabetes of adults (LADA). Here is what Wikipedia says about LADA. Officially it can nevertheless be mentioned to be kind 1 diabetes. This really is a smaller post of mine, however you are able to observe the 2 videos below and you can also buy Dr. Cousen's book on curing diabetes.
The Body Mass Index is based on a person's fat inside proportion to their height plus age. The National Institute of Health provides a convenient online bmi calculator females that automatically computes the value based found on the entered height plus weight. Results under 18.5 get into the "underweight" range, that may cause many wellness risks including malnurition and emaciation.
BMI is selected worldwide to determine if an individual is obese. Since BMI is chosen only for a screening tool, anybody whom says which you are at health risk before performing any different tests is lying. Doctors may screen individuals for wellness risks after seeing which their BMI is above normal.
There were 4 others that had kind 2 diabetes plus one with type 1 diabetes. After a month, the one with kind 1 diabetes was able to drop his insulin from 70 units right down to 5 units a day. With the 4 others, they were able to get off all of their medicine and have a normal blood glucose level meaning which they were cured.
In terms of its dodgy mathematics, the BMI is borderline junk research. However inside this regard, BMI is less egregious than Global Warming 'studies', inside that many of the big-name 'researchers' cherry-pick, hide, or even fabricate data. Click found on the link for my lengthy hub on the topic.