Binomial distribution
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| Probability mass function | |
| Cumulative distribution function | |
| Parameters | <math>n \geq 0</math> number of trials (integer) <math>0\leq p \leq 1</math> success probability (real) |
|---|---|
| Support | <math>k \in \{0,\dots,n\}\!</math> |
| Probability mass function (pmf) | <math>{n\choose k} p^k (1-p)^{n-k} \!</math> |
| Cumulative distribution function (cdf) | <math>I_{1-p}(n-\lfloor k\rfloor, 1+\lfloor k\rfloor) \!</math> |
| Mean | <math>np\!</math> |
| Median | one of <math>\{\lfloor np\rfloor-1, \lfloor np\rfloor, \lfloor np\rfloor+1\}</math> |
| Mode | <math>\lfloor (n+1)\,p\rfloor\!</math> |
| Variance | <math>np(1-p)\!</math> |
| Skewness | <math>\frac{1-2p}{\sqrt{np(1-p) |
| Excess Kurtosis | {{{kurtosis}}} |
| Entropy | {{{entropy}}} |
| mgf | {{{mgf}}} |
| Char. func. | {{{char}}} |
kurtosis =<math>\frac{1-6p(1-p)}{np(1-p)}\!</math>|
entropy =<math> \frac{1}{2} \ln \left( 2 \pi n e p (1-p) \right) + O \left( \frac{1}{n} \right) </math>|
mgf =<math>(1-p + pe^t)^n \!</math>|
char =<math>(1-p + pe^{it})^n \!</math>|
}}
In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial. In fact, when n = 1, then the binomial distribution is the Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.
Contents |
[edit] Example
A typical example is the following: assume 5% of the population is green-eyed. You pick 500 people randomly. The number of green-eyed people you pick is a random variable X which follows a binomial distribution with n = 500 and p = 0.05 (when picking the people with replacement).
[edit] Specification
[edit] Probability mass function
In general, if the random variable X follows the binomial distribution with parameters n and p, we write X ~ B(n, p). The probability of getting exactly k successes is given by the probability mass function:
- <math>f(k;n,p)={n\choose k}p^k(1-p)^{n-k}\,</math>
for k=0,1,2,...,n and where
- <math>{n\choose k}=\frac{n!}{k!(n-k)!}</math>
is the binomial coefficient (hence the name of the distribution) "n choose k" (also denoted C(n, k) or nCk). The formula can be understood as follows: we want k successes (pk) and n − k failures (1 − p)n − k. However, the k successes can occur anywhere among the n trials, and there are C(n, k) different ways of distributing k successes in a sequence of n trials.
In creating reference tables for binomial distribution probability, usually the table is filled in up to n/2 values. This is because for k > n/2, the probability can be calculated by its complement as
- <math>f(k;n,p)=f(n-k;n,1-p).\,\!</math>
So, one must look to a different k and a different p (the binomial is not symmetrical in general).
[edit] Cumulative distribution function
The cumulative distribution function can be expressed in terms of the regularized incomplete beta function, as follows:
- <math> F(k;n,p) = \Pr(X \le k) = I_{1-p}(n-k, k+1) \!</math>
provided k is an integer and 0 ≤ k ≤ n. If x is not necessarily an integer or not necessarily positive, one can express it thus:
- <math>F(x;n,p) = \Pr(X \le x) = \sum_{j=0}^{\operatorname{Floor}(x)} {n\choose j}p^j(1-p)^{n-j}</math>
For k ≤ np, upper bounds for the lower tail of the distribution function can be derived. In particular, Hoeffding's inequality yields the bound
- <math> F(k;n,p) \leq \exp\left(-2 \frac{(np-k)^2}{n}\right), \!</math>
and Chernoff's inequality can be used to derive the bound
- <math> F(k;n,p) \leq \exp\left(-\frac{1}{2\,p} \frac{(np-k)^2}{n}\right). \!</math>
[edit] Mean, standard deviation, and mode
If X ~ B(n, p) (that is, X is a binomially distributed random variable), then the expected value of X is
- <math>\operatorname{E}(X)=np\,\!</math>
and the variance is
- <math>\operatorname{Var}(X)=np(1-p).\,\!</math>
This fact is easily proven as follows. Suppose first that we have exactly one Bernoulli trial. We have two possible outcomes, 1 and 0, with the first having probability p and the second having probability 1 − p; the mean for this trial is given by μ = p. Using the definition of variance, we have
- <math>\sigma^2= \left(1 - p\right)^2p + (0-p)^2(1 - p) = p(1-p).</math>
Now suppose that we want the variance for n such trials (i.e. for the general binomial distribution). Since the trials are independent, we may add the variances for each trial, giving
- <math>\sigma^2_n = \sum_{k=1}^n \sigma^2 = np(1 - p). \quad \Box</math>
The most likely value or mode of X is given by the largest integer less than or equal to (n + 1)p; if m = (n + 1)p is itself an integer, then m − 1 and m are both modes.
[edit] Relations to other distributions
[edit] Sums of binomials
If X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables, then X + Y is again a binomial variable; its distribution is
- <math>X+Y \sim B(n+m, p).\,</math>
[edit] Normal approximation
If n is large enough, the skew of the distribution is not too great, and a suitable continuity correction is used, then an excellent approximation to B(n, p) is given by the normal distribution
- <math> \operatorname{N}(np, np(1-p)).\,\!</math>
Various rules of thumb may be used to decide whether n is large enough. One rule is that both np and n(1 − p) must be greater than 5. However, the specific number varies from source to source, and depends on how good an approximation one wants; some sources give 10. Another commonly used rule holds that the above normal approximation is appropriate only if
- <math>\mu \pm 3 \sigma = np \pm 3 \sqrt{np(1-p)} \in [0,n].</math>
The following is an example of applying a continuity correction: Suppose one wishes to calculate Pr(X ≤ 8) for a binomial random variable X. If Y has a distribution given by the normal approximation, then Pr(X ≤ 8) is approximated by Pr(Y ≤ 8.5). The addition of 0.5 is the continuity correction. Warning: The normal approximation gives inaccurate results unless a continuity correction is used.
This approximation is a huge time-saver (exact calculations with large n are very onerous); historically, it was the first use of the normal distribution, introduced in Abraham de Moivre's book The Doctrine of Chances in 1733. Nowadays, it can be seen as a consequence of the central limit theorem since B(n, p) is a sum of n independent, identically distributed 0-1 indicator variables.
For example, suppose you randomly sample n people out of a large population and ask them whether they agree with a certain statement. The proportion of people who agree will of course depend on the sample. If you sampled groups of n people repeatedly and truly randomly, the proportions would follow an approximate normal distribution with mean equal to the true proportion p of agreement in the population and with standard deviation σ = (p(1 − p)/n)1/2. Large sample sizes n are good because the standard deviation gets smaller, which allows a more precise estimate of the unknown parameter p.
[edit] Poisson approximation
The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product np remains fixed. Therefore the Poisson distribution with parameter λ = np can be used as an approximation to B(n, p) of the binomial distribution if n is sufficiently large and p is sufficiently small. According to one rule of thumb, this approximation is good if n ≥ 20 and p ≤ 0.05, and also if n ≥ 100 and np ≤ 10.<ref>NIST/SEMATECH, '6.3.3.1. Counts Control Charts', e-Handbook of Statistical Methods, <http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc331.htm> [accessed 25 October 2006]</ref>
[edit] Limits of binomial distributions
- As n approaches ∞ and p approaches 0 while np remains fixed at λ > 0 or at least np approaches λ > 0, then the Binomial(n, p) distribution approaches the Poisson distribution with expected value λ.
- As n approaches ∞ while p remains fixed, the distribution of
- <math>{X-np \over \sqrt{np(1-p)\ }}</math>
- approaches the normal distribution with expected value 0 and variance 1.
[edit] References
<references/>
- Abdi, H. "[1] ((2007). Binomial Distribution: Binomial and Sign Tests.. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.".
- Luc Devroye, Non-Uniform Random Variate Generation, New York: Springer-Verlag, 1986. See especially Chapter X, Discrete Univariate Distributions.
- Voratas Kachitvichyanukul and Bruce W. Schmeiser, Binomial random variate generation, Communications of the ACM 31(2):216–222, February 1988. DOI:10.1145/42372.42381
- Cheatam & Steele, "Uniform Distributive Norms", Los Angeles: Time-Warner, 1998.
[edit] See also
- Bean machine / Galton board
- Beta distribution
- Multinomial distribution
- Negative binomial distribution
- Poisson distribution
- Hypergeometric distribution
[edit] External links
- Binomial Probability Distribution Calculator
- Binomial Probabilities Simple Explanationcs:Binomické rozdělení
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