Difference between revisions of "Binomial distribution"
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Consider an event—such as a coin coming down heads—that can be true or false in each trial—or each toss—with probability «p» -- it has a [[Bernoulli]] distribution. A binomial distribution describes the number of times an event is true -- e.g., the coin is heads -- in «n» independent trials—or tosses—where the event occurs with probability «p» on each trial. | Consider an event—such as a coin coming down heads—that can be true or false in each trial—or each toss—with probability «p» -- it has a [[Bernoulli]] distribution. A binomial distribution describes the number of times an event is true -- e.g., the coin is heads -- in «n» independent trials—or tosses—where the event occurs with probability «p» on each trial. | ||
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| + | <center>[[image:BinomialDistribution.png]]</center> | ||
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| + | == Functions == | ||
| + | === Parameters === | ||
| + | The binomial distribution is described by | ||
| + | * «n»: The number of trials | ||
| + | * «p»: The probability of success in each trial | ||
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| + | === Binomial(n, p) === | ||
| + | The number of successes for a quantity described by a binomial distribution. This is the Poisson [[:category|Distribution function]]. Use this to describe an uncertain variable that represents the number of successes in a repeated trial. | ||
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| + | === <div id="ProbBinomial">Prob{{Release||5.1|_}}Binomial(k, n, p)</div> === | ||
| + | {{Release||5.1|To use, add the [[Distribution Densities Library]] to your model.}} | ||
The Binomial distribution is a non-negative discrete distribution where the probability of outcome ''k'' is given by | The Binomial distribution is a non-negative discrete distribution where the probability of outcome ''k'' is given by | ||
:<math>P_{n,p}(k) = \left(\begin{array}{c}n\\k\end{array}\right) p^k (1-p)^{n-k}</math> | :<math>P_{n,p}(k) = \left(\begin{array}{c}n\\k\end{array}\right) p^k (1-p)^{n-k}</math> | ||
| − | + | === <div id="CumBinomial">CumBinomial(k, n, p)</div> === | |
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| + | The [[:category:Cumulative distribution functions|analytic cumulative probability function]] for a Poisson distribution. Returns the probability that the actual number of successes is less than or equal to «k». | ||
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| + | === <div id="CumBinomialInv">CumBinomialInv(u, n, p)</div> === | ||
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| + | The [[:category:Inverse cumulative probability functions|analytic inverse cumulative probability function]] for a Poisson(n,p) distribution, also known as the ''quantile function''. For 0<=u<1, this returns the number of successes, k, so that the probability of the outcome of «n» trials having «k» or fewer successes is less than or equal to «k». | ||
| + | == Statistics == | ||
The distribution has a [[Mean]] of <code>n*p</code> and a [[Variance]] of <code>n*p*(1 - p)</code>. | The distribution has a [[Mean]] of <code>n*p</code> and a [[Variance]] of <code>n*p*(1 - p)</code>. | ||
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== See Also == | == See Also == | ||
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* [[Multinomial]] -- A generalization of Binomial in which more than two outcomes are possible. | * [[Multinomial]] -- A generalization of Binomial in which more than two outcomes are possible. | ||
* [[NegativeBinomial]] -- the two other common discrete distributions on the non-negative integers | * [[NegativeBinomial]] -- the two other common discrete distributions on the non-negative integers | ||
| − | * [[Poisson]] | + | * [[Poisson distribution]] |
| − | * [[Normal]] | + | * [[Normal distribution]] |
* [[Parametric discrete distributions]] | * [[Parametric discrete distributions]] | ||
* [[Distribution Densities Library]] | * [[Distribution Densities Library]] | ||
Revision as of 22:50, 7 December 2018
Consider an event—such as a coin coming down heads—that can be true or false in each trial—or each toss—with probability «p» -- it has a Bernoulli distribution. A binomial distribution describes the number of times an event is true -- e.g., the coin is heads -- in «n» independent trials—or tosses—where the event occurs with probability «p» on each trial.
Functions
Parameters
The binomial distribution is described by
- «n»: The number of trials
- «p»: The probability of success in each trial
Binomial(n, p)
The number of successes for a quantity described by a binomial distribution. This is the Poisson Distribution function. Use this to describe an uncertain variable that represents the number of successes in a repeated trial.
Prob_Binomial(k, n, p)
To use, add the Distribution Densities Library to your model.
The Binomial distribution is a non-negative discrete distribution where the probability of outcome k is given by
- [math]\displaystyle{ P_{n,p}(k) = \left(\begin{array}{c}n\\k\end{array}\right) p^k (1-p)^{n-k} }[/math]
CumBinomial(k, n, p)
The analytic cumulative probability function for a Poisson distribution. Returns the probability that the actual number of successes is less than or equal to «k».
CumBinomialInv(u, n, p)
The analytic inverse cumulative probability function for a Poisson(n,p) distribution, also known as the quantile function. For 0<=u<1, this returns the number of successes, k, so that the probability of the outcome of «n» trials having «k» or fewer successes is less than or equal to «k».
Statistics
The distribution has a Mean of n*p and a Variance of n*p*(1 - p).
See Also
- Multinomial -- A generalization of Binomial in which more than two outcomes are possible.
- NegativeBinomial -- the two other common discrete distributions on the non-negative integers
- Poisson distribution
- Normal distribution
- Parametric discrete distributions
- Distribution Densities Library
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