Erlang distribution - Revision history

en>IkamusumeFan: I have created the ad hoc pdf and cdf for Erlang. The previous ones were essentially Gamma, which exceeded Erlang.

2014-12-08T23:28:45Z

I have created the ad hoc pdf and cdf for Erlang. The previous ones were essentially Gamma, which exceeded Erlang.

← Older revision		Revision as of 01:28, 9 December 2014
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	~~{{refimprove\|date=May 2010}}~~		I'm Sherry and I live with my husband and our three children in Eberswalde, in the BB south part. My hobbies are Gardening, Martial arts and Baseball.<br><br>Feel free to surf to my website ... [http://fifa15-coingenerator.com/ FIFA coin generator]
	~~{{Probability distribution\|~~
	~~name =Bernoulli\|~~
	~~type =mass\|~~
	~~pdf_image =\|~~
	~~cdf_image =\|~~
	~~parameters =<math>0<p<1, p\in\R</math>\|~~
	~~support =<math>k \in \{0,1\}\,</math>\|~~
	~~pdf =<math>~~
	~~\begin{cases}~~
	~~q=(1-p) & \text{for }k=0 \\ p & \text{for }k=1~~
	~~\end{cases}~~
	~~</math>\|~~
	~~cdf =<math>~~
	~~\begin{cases}~~
	~~0 & \text{for }k<0 \\ q & \text{for }0\leq k<1 \\ 1 & \text{for }k\geq 1~~
	~~\end{cases}~~
	~~</math>\|~~
	~~mean =<math>p\,</math>\|~~
	~~median =<math>\begin{cases}~~
	~~0 & \text{if } q > p\\~~
	~~0.5 & \text{if } q=p\\~~
	~~1 & \text{if } q<p~~
	~~\end{cases}</math>\|~~
	~~mode =<math>\begin{cases}~~
	~~0 & \text{if } q > p\\~~
	~~0, 1 & \text{if } q=p\\~~
	~~1 & \text{if } q < p~~
	~~\end{cases}</math>\|~~
	~~variance =<math>p(1-p)\,</math>\|~~
	~~skewness =<math>\frac{q-p}{\sqrt{pq}}</math>\|~~
	~~kurtosis =<math>\frac{1-6pq}{pq}</math>\|~~
	~~entropy =<math>-q\ln(q)-p\ln(p)\,</math>\|~~
	~~mgf =<math>q+pe^t\,</math>\|~~
	~~char =<math>q+pe^{it}\,</math>\|~~
	~~pgf =<math>q+pz\,</math>\|~~
	~~fisher = <math> \frac{1}{p(1-p)} </math>\|~~
	}}

	~~In [[probability theory]] and [[statistics]], the~~ '''Bernoulli distribution''', named after Swiss scientist [[Jacob Bernoulli]], is a [[discrete probability distribution\|discrete]] [[probability distribution]], which takes value 1 with success probability <math>p</math> and ~~value 0~~ with ~~failure probability <math>q=1-p</math>.~~

	~~==Properties==~~
	~~If <math>X</math> is a random variable with this distribution, we have:~~

	~~:<math> \Pr(X=1) = 1 - \Pr(X=0) = 1 - q = p.\!</math>~~

	~~A classical example of a Bernoulli experiment is a single toss of a coin. The coin might come up heads with probability <math>p</math>~~ and ~~tails with probability <math>1-p</math>. The experiment is called fair if <math>p=0.5</math>~~, ~~indicating the origin of the terminology~~ in ~~betting (the bet is fair if both possible outcomes have~~ the ~~same probability)~~.

	~~The [[probability mass function]] <math>f</math> of this distribution is~~

	~~: <math> f(k;p) = \begin{cases} p & \text{if }k=1, \\[6pt]~~
	~~1-p & \text {if }k=0.\end{cases}</math>~~

	~~This can also be expressed as~~

	~~:<math>f(k;p) = p^k (1-p)^{1-k}\!\quad \text{for }k\in\{0,1\}.</math>~~

	~~The [[expected value]] of a Bernoulli random variable <math>X</math> is <math>E\left(X\right)=p</math>~~, and ~~its [[variance]] is~~

	~~:<math>\textrm{Var}\left(X\right)=p\left(1-p\right)~~.\,<~~/math~~>

	~~Bernoulli distribution is a special case of the [[Binomial distribution]] with~~ <~~math~~>~~n = 1</math>.<ref name="McCullagh1989Ch422">[[#McCullagh1989\|McCullagh and Nelder (1989)]], Section 4.2.2.</ref>~~

	~~The [[kurtosis]] goes~~ to ~~infinity for high and low values of <math>p</math>, but for <math>p=1/2</math> the Bernoulli distribution has a lower excess kurtosis than any other probability distribution, namely −2.~~

	~~The Bernoulli distributions for <math>0 \le p \le 1</math> form an [[exponential family]].~~

	~~The [[maximum likelihood estimator]] of <math>p</math> based on a random sample is the [[sample mean]].~~

	~~==Related distributions==~~
	*If <math>X_1,\dots,X_n</math> are independent, identically distributed ([[i.i.d.]]) random variables, all Bernoulli distributed with success probability ''p'', then <math>Y = \sum_{k=1}^n X_k \sim \mathrm{Binomial}(n,p)</math> ([[binomial distribution]]). The Bernoulli distribution is simply <math>\mathrm{Binomial}(1,p)</math>.
	*The [[categorical distribution]] is the generalization of the Bernoulli distribution for variables with any constant number of discrete values.
	*The [[Beta distribution]] is the [[conjugate prior]] of the Bernoulli distribution.
	*The [[geometric distribution]] models the number of independent and identical Bernoulli trials needed to ~~get one success.~~

	~~==See also==~~
	*[[Bernoulli process]]
	*[[Bernoulli sampling]]
	*[[Bernoulli trial]]
	*[[Binary entropy function]]

	~~== Notes ==~~
	~~{{reflist}}~~

	~~==References==~~

	* {{cite book \| last = McCullagh \| first = Peter \| authorlink= Peter McCullagh \| coauthors = [[John Nelder\|Nelder, John]] \| title = Generalized Linear Models, Second Edition \| publisher = Boca Raton: Chapman and Hall/CRC \| year = 1989 \| isbn = 0-412-31760-5 \|ref=McCullagh1989}}

	*Johnson, N.L., Kotz, S., Kemp A. ~~(1993) Univariate Discrete Distributions (2nd Edition)~~. ~~Wiley~~. ~~ISBN 0-471-54897-9~~

	~~==External links==~~
	*{{springer\|title=Binomial distribution\|id=p/b016420}}
	*{{MathWorld\|title=Bernoulli Distribution\|urlname=BernoulliDistribution}}
	* Interactive graphic: [http://~~www.math.wm~~.~~edu~~/~~~leemis/chart/UDR/UDR.html Univariate Distribution Relationships]~~

	~~{{ProbDistributions\|discrete-finite}}~~
	~~{{Common univariate probability distributions}}~~

	~~{{DEFAULTSORT:Bernoulli Distribution}}~~
	~~[[Category:Discrete distributions]]~~
	~~[[Category:Distributions with conjugate priors]]~~
	~~[[Category:Exponential family distributions]]~~
	~~[[Category:Probability distributions]~~]

en>BeyondNormality: /* External links */

2013-12-10T03:39:49Z

External links

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76.119.85.156: /* Related distributions */

2012-08-30T11:33:42Z

Related distributions

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