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Probability mass function  
Cumulative distribution function  
Notation  B(n, p) 

Parameters  n ∈ N_{0} — number of trials p ∈ [0,1] — success probability in each trial 
Support  k ∈ { 0, …, n } — number of successes 
pmf  
CDF  
Mean  np 
Median  ⌊np⌋ or ⌈np⌉ 
Mode  ⌊(n + 1)p⌋ or ⌊(n + 1)p⌋ − 1 
Variance  np(1 − p) 
Skewness  
Ex. kurtosis  
Entropy  
MGF  
CF  
PGF  
Fisher information  (continuous parameter only) 
Probability mass function  
Cumulative distribution function  
Notation  B(n, p) 

Parameters  n ∈ N_{0} — number of trials p ∈ [0,1] — success probability in each trial 
Support  k ∈ { 0, …, n } — number of successes 
pmf  
CDF  
Mean  np 
Median  ⌊np⌋ or ⌈np⌉ 
Mode  ⌊(n + 1)p⌋ or ⌊(n + 1)p⌋ − 1 
Variance  np(1 − p) 
Skewness  
Ex. kurtosis  
Entropy  
MGF  
CF  
PGF  
Fisher information  (continuous parameter only) 
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; when n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution is a good approximation, and widely used.
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 in n trials is given by the probability mass function:
for k = 0, 1, 2, ..., n, where
is the binomial coefficient, hence the name of the distribution. The formula can be understood as follows: we want k successes (p^{k}) and n − k failures (1 − p)^{n − k}. However, the k successes can occur anywhere among the n trials, and there are 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
Looking at the expression ƒ(k, n, p) as a function of k, there is a k value that maximizes it. This k value can be found by calculating
and comparing it to 1. There is always an integer M that satisfies
ƒ(k, n, p) is monotone increasing for k < M and monotone decreasing for k > M, with the exception of the case where (n + 1)p is an integer. In this case, there are two values for which ƒ is maximal: (n + 1)p and (n + 1)p − 1. M is the most probable (most likely) outcome of the Bernoulli trials and is called the mode. Note that the probability of it occurring can be fairly small.
The cumulative distribution function can be expressed as:
where is the "floor" under k, i.e. the greatest integer less than or equal to k.
It can also be represented in terms of the regularized incomplete beta function, as follows:^{[1]}
Some closedform bounds for the cumulative distribution function are given below.
Suppose a biased coin comes up heads with probability 0.3 when tossed. What is the probability of achieving 0, 1,..., 6 heads after six tosses?
If X ~ B(n, p), that is, X is a binomially distributed random variable, n being the total number of experiments and p the probability of each experiment yielding a successful result, then the expected value of X is
(For example, if n=100, and p=25/100, then 25% of the results are likely to be successful. It is expected that 25 results are successful even though it is not certain.)
and the variance
Usually the mode of a binomial B(n, p) distribution is equal to , where is the floor function. However when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes: (n + 1)p and (n + 1)p − 1. When p is equal to 0 or 1, the mode will be 0 and n correspondingly. These cases can be summarized as follows:
In general, there is no single formula to find the median for a binomial distribution, and it may even be nonunique. However several special results have been established:
If two binomially distributed random variables X and Y are observed together, estimating their covariance can be useful. Using the definition of covariance, in the case n = 1 (thus being Bernoulli trials) we have
The first term is nonzero only when both X and Y are one, and μ_{X} and μ_{Y} are equal to the two probabilities. Defining p_{B} as the probability of both happening at the same time, this gives
and for n independent pairwise trials
If X and Y are the same variable, this reduces to the variance formula given above.
If X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables with the same probability p, then X + Y is again a binomial variable; its distribution is^{[citation needed]}
If X ~ B(n, p) and, conditional on X, Y ~ B(X, q), then Y is a simple binomial variable with distribution^{[citation needed]}
The Bernoulli distribution is a special case of the binomial distribution, where n = 1. Symbolically, X ~ B(1, p) has the same meaning as X ~ Bern(p). Conversely, any binomial distribution, B(n, p), is the distribution of the sum of n Bernoulli trials, Bern(p), each with the same probability p.^{[citation needed]}
The binomial distribution is a special case of the Poisson binomial distribution, which is a sum of n independent nonidentical Bernoulli trials Bern(p_{i}).^{[citation needed]} If X has the Poisson binomial distribution with p_{1} = … = p_{n} =p then X ~ B(n, p).
If n is large enough, then the skew of the distribution is not too great. In this case a reasonable approximation to B(n, p) is given by the normal distribution
and this basic approximation can be improved in a simple way by using a suitable continuity correction. The basic approximation generally improves as n increases (at least 20) and is better when p is not near to 0 or 1.^{[7]} Various rules of thumb may be used to decide whether n is large enough, and p is far enough from the extremes of zero or one:
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; the uncorrected normal approximation gives considerably less accurate results.
This approximation, known as de Moivre–Laplace theorem, is a huge timesaver when undertaking calculations by hand (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 1738. 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 Bernoulli variables with parameter p. This fact is the basis of a hypothesis test, a "proportion ztest", for the value of p using x/n, the sample proportion and estimator of p, in a common test statistic.^{[8]}
For example, suppose one randomly samples 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 groups of n people were sampled 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, as a proportion of the expected value, gets smaller, which allows a more precise estimate of the unknown parameter p.
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 two rules of thumb, this approximation is good if n ≥ 20 and p ≤ 0.05, or if n ≥ 100 and np ≤ 10.^{[9]}
Beta distributions provide a family of conjugate prior probability distributions for binomial distributions in Bayesian inference. The domain of the beta distribution can be viewed as a probability, and in fact the beta distribution is often used to describe the distribution of a probability value p:^{[10]}
Even for quite large values of n, the actual distribution of the mean is significantly nonnormal.^{[11]} Because of this problem several methods to estimate confidence intervals have been proposed.
Let n_{1} be the number of successes out of n, the total number of trials, and let
be the proportion of successes. Let z_{α/2} be the 100(1 − α/2)th percentile of the standard normal distribution.
The exact (ClopperPearson) method is the most conservative.^{[11]} The Wald method although commonly recommended in the text books is the most biased.^{[clarification needed]}
Methods for random number generation where the marginal distribution is a binomial distribution are wellestablished.^{[15]}^{[16]}
One way to generate random samples from a binomial distribution is to use an inversion algorithm. To do so, one must calculate the probability that P(X=k) for all values k from 0 through n. (These probabilities should sum to a value close to one, in order to encompass the entire sample space.) Then by using a Linear congruential generator to generate samples uniform between 0 and 1, one can transform the calculated samples U[0,1] into discrete numbers by using the probabilities calculated in step one.
For k ≤ np, upper bounds for the lower tail of the distribution function can be derived. In particular, Hoeffding's inequality yields the bound
and Chernoff's inequality can be used to derive the bound
Moreover, these bounds are reasonably tight when p = 1/2, since the following expression holds for all k ≥ 3n/8^{[17]}
However, the bounds do not work well for extreme values of p. In particular, as p 1, value F(k;n,p) goes to zero (for fixed k, n with k<n) while the upper bound above goes to a positive constant. In this case a better bound is given by ^{[18]}
where H(a, p) is the relative entropy between a pcoin and an acoin:
Asymptotically, this bound is reasonably tight; see ^{[18]} for details. An equivalent formulation of the bound is
