Binomial distribution

"Binomial model" redirects here. For the binomial model in options pricing, see Binomial options pricing model.

See also: Negative binomial distribution

Probability mass function
Cumulative distribution function
Notation	B(n, p)
Parameters	n ∈ N0 — number of trials p ∈ [0,1] — success probability in each trial
Support	k ∈ { 0, …, n } — number of successes
PMF	$\textstyle {n \choose k}\,p^{k}(1-p)^{n-k}$
CDF	$\textstyle I_{1-p}(n-k,1+k)$
Mean	np
Median	⌊np⌋ or ⌈np⌉
Mode	⌊(n + 1)p⌋ or ⌊(n + 1)p⌋ − 1
Variance	np(1 − p)
Skewness	${\frac {1-2p}{\sqrt {np(1-p)}}}$
Excess kurtosis	${\frac {1-6p(1-p)}{np(1-p)}}$
Entropy	${\frac {1}{2}}\log _{2}{\big (}2\pi e\,np(1-p){\big )}+O\left({\frac {1}{n}}\right)$
MGF	$(1-p+pe^{t})^{n}\!$
CF	$(1-p+pe^{it})^{n}\!$
PGF	$G(z)=\left[(1-p)+pz\right]^{n}.$

Binomial distribution for $n=20$
$p=0.1$ (blue), $p=0.5$ (green) and $p=0.8$ (red)

Binomial distribution for $p=0.5$
with $n$ and $k$ as in Pascal's triangle

The probability that a ball in a Galton box with 8 layers ($n=8$) ends up in the central bin ($k=4$) is $70/256$.

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.

Specification

Probability mass function

In general, if the random variable K follows the binomial distribution with parameters n and p, we write K ~ B(n, p). The probability of getting exactly k successes in n trials is given by the probability mass function:

$f(k;n,p)=\Pr(K=k)={n \choose k}p^{k}(1-p)^{n-k}$

for k = 0, 1, 2, ..., n, where

${n \choose k}={\frac {n!}{k!(n-k)!}}$

is the binomial coefficient (hence the name of the distribution) "n choose k", also denoted C(n, k),  _nC_k, or ⁿC_k. 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 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

$f(k,n,p)=f(n-k;n,1-p).\,$

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

${\frac {f(k+1,n,p)}{f(k,n,p)}}={\frac {(n-k)p}{(k+1)(1-p)}}$

and comparing it to 1. There is always an integer M that satisfies

$(n+1)p-1<M\leq (n+1)p.\,$

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.

Cumulative distribution function

The cumulative distribution function can be expressed as:

$F(x;n,p)=\Pr(X\leq x)=\sum _{i=0}^{\lfloor x\rfloor }{n \choose i}p^{i}(1-p)^{n-i}$

where $\scriptstyle \lfloor x\rfloor \,$ is the "floor" under x, i.e. the greatest integer less than or equal to x.

It can also be represented in terms of the regularized incomplete beta function, as follows:

${\begin{aligned}F(k;n,p)&=\Pr(X\leq k)=I_{1-p}(n-k,k+1)\\&=(n-k){n \choose k}\int _{0}^{1-p}t^{n-k-1}(1-t)^{k}\,dt.\end{aligned}}$

For k ≤ np, upper bounds for the lower tail of the distribution function can be derived. In particular, Hoeffding's inequality yields the bound

$F(k;n,p)\leq {\frac {1}{2}}\exp \left(-2{\frac {(np-k)^{2}}{n}}\right),\!$

and Chernoff's inequality can be used to derive the bound

$F(k;n,p)\leq \exp \left(-{\frac {1}{2\,p}}{\frac {(np-k)^{2}}{n}}\right).\!$

Moreover, these bounds are reasonably tight when p = 1/2, since the following expression holds for all k ≥ 3n/8

$F(k;n,1/2)\geq {\frac {1}{15}}\exp \left(-{\frac {16(n/2-k)^{2}}{n}}\right).\!$

Mean and variance

If X ~ B(n, p) (that is, X is a binomially distributed random variable), then the expected value of X is

$\operatorname {E} [X]=np$

and the variance is

$\operatorname {Var} [X]=np(1-p).$

This fact is easily proven as follows. Suppose first that we have a single Bernoulli trial. There are two possible outcomes: 1 and 0, the first occurring with probability p and the second having probability 1 − p. The expected value in this trial will be equal to μ = 1 · p + 0 · (1−p) = p. The variance in this trial is calculated similarly: σ² = (1−p)²·p + (0−p)²·(1−p) = p(1 − p).

The generic binomial distribution is a sum of n independent Bernoulli trials. The mean and the variance of such distributions are equal to the sums of means and variances of each individual trial:

$\mu _{n}=\sum _{k=1}^{n}\mu =np,\qquad \sigma _{n}^{2}=\sum _{k=1}^{n}\sigma ^{2}=np(1-p).$

Mode and median

Usually the mode of a binomial B(n, p) distribution is equal to $\lfloor (n+1)p\rfloor$, where $\lfloor \cdot \rfloor$ 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:

${\text{mode}}={\begin{cases}\lfloor (n+1)\,p\rfloor &{\text{if }}(n+1)p{\text{ is 0 or a noninteger}},\\(n+1)\,p\ {\text{ and }}\ (n+1)\,p-1&{\text{if }}(n+1)p\in \{1,\dots ,n\},\\n&{\text{if }}(n+1)p=n+1.\end{cases}}$

In general, there is no single formula to find the median for a binomial distribution, and it may even be non-unique. However several special results have been established:

• If np is an integer, then the mean, median, and mode coincide and equal np.
• Any median m must lie within the interval ⌊np⌋ ≤ m ≤ ⌈np⌉.
• A median m cannot lie too far away from the mean: |m − np| ≤ min{ ln 2, max{p, 1 − p} }.
• The median is unique and equal to m = round(np) in cases when either p ≤ 1 − ln 2 or p ≥ ln 2 or |m − np| ≤ min{p, 1 − p} (except for the case when p = ½ and n is odd).
• When p = 1/2 and n is odd, any number m in the interval ½(n − 1) ≤ m ≤ ½(n + 1) is a median of the binomial distribution. If p = 1/2 and n is even, then m = n/2 is the unique median.

Covariance between two binomials

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

$\operatorname {Cov} (X,Y)=\operatorname {E} (XY)-\mu _{X}\mu _{Y}.$

The first term is non-zero 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

$\operatorname {Cov} (X,Y)=p_{B}-p_{X}p_{Y},\,$

and for n such trials again due to independence

$\operatorname {Cov} (X,Y)_{n}=n(p_{B}-p_{X}p_{Y}).\,$

If X and Y are the same variable, this reduces to the variance formula given above.

Relationship to other distributions

Sums of binomials

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

$X+Y\sim B(n+m,p).\,$

Conditional binomials

If X ~ B(n, p) and, conditional on X, Y ~ B(X, q), then Y is a simple binomial variable with distribution

$Y\sim B(n,pq).\,$

Bernoulli distribution

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 sum of n independent Bernoulli trials, Bern(p), each with the same probability p.

Poisson binomial distribution

The binomial distribution is a special case of the Poisson binomial distribution, which is a sum of n independent non-identical Bernoulli trials Bern(p_i). If X has the Poisson binomial distribution with p₁ = … = p_n =p then X ~ B(n, p).

Normal approximation

Binomial PDF and normal approximation for n = 6 and p = 0.5

If n is large enough, then the skew of the distribution is not too great. In this case, if a suitable continuity correction is used, then an excellent approximation to B(n, p) is given by the normal distribution

${\mathcal {N}}(np,\,np(1-p)).$

The approximation generally improves as n increases (at least 20) and is better when p is not near to 0 or 1. 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:

• One rule is that both x=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 which gives virtually the same results as the following rule for large n until n is very large (ex: x=11, n=7752).

• That rule is that for n > 5 the normal approximation is adequate if

$|(1/{\sqrt {n}})({\sqrt {(1-p)/p}}-{\sqrt {p/(1-p)}})|<0.3$

• Another commonly used rule holds that the normal approximation is appropriate only if everything within 3 standard deviations of its mean is within the range of possible values, that is if

$\mu \pm 3\sigma =np\pm 3{\sqrt {np(1-p)}}\in [0,n].\,$

• Also as the approximation generally improves, it can be shown that the inflection points occur at

$np\pm {\sqrt {np(1-p)}}\,$

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 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 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 z-test," for the value of p using x/n, the sample proportion and estimator of p, in a common test statistic.

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, as a proportion of the expected value, gets smaller, which allows a more precise estimate of the unknown parameter p.

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 two rules of thumb, this approximation is good if n ≥ 20 and p ≤ 0.05, or if n ≥ 100 and np ≤ 10.

Limits

• Poisson limit theorem: 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.

• de Moivre–Laplace theorem: As n approaches ∞ while p remains fixed, the distribution of

${X-np \over {\sqrt {np(1-p)\ }}}$

approaches the normal distribution with expected value 0 and variance 1. This result is sometimes loosely stated by saying that the distribution of X approaches the normal distribution with expected value np and variance np(1 − p). That loose statement cannot be taken literally because the thing asserted to be approached actually depends on the value of n, and n is approaching infinity. This result is a specific case of the Central Limit Theorem.

Generating binomial random variates

• Luc Devroye, Non-Uniform Random Variate Generation, New York: Springer-Verlag, 1986. See especially Chapter X, Discrete Univariate Distributions.

Examples

An elementary example is this: roll a standard die ten times and count the number of fours. The distribution of this random number is a binomial distribution with n = 10 and p = 1/6.

Symmetric binomial distribution (p = 0.5)

This example illustrates the central limit theorem in the example of fair coin tosses. A coin is fair if it shows heads and tails with equal probability p = 1/2. If we toss a fair coin n times, the number k of times the coin shows 'heads' is a random variable that is binomially distributed. The first picture below shows the binomial distribution for several values of n as a function of k.

• 
  Binomial distributions with p = 0.5 and n = 4,16,64
• 
  Moved mean to zero
• 
  Rescaled with standard deviation

Rescaled and shifted binomial distributions with p = 0.5 for n = 4, 6, 8, 12, 16, 23, 32, 46

The same data as above on a logarithmic scale

These distributions are reflection symmetric with respect to the line k = n/2, that is, the probability mass function satisfies ƒ(k; n, 1/2) = ƒ(n − k; n, 1/2). In the middle picture, the distributions were shifted so that their mean is zero.

The width of the distribution is proportional to the standard deviation $\sigma ={\tfrac {1}{2}}{\sqrt {n}}$. The value of the shifted functions at $k=0$ is their respective maximum and proportional to $1/\sigma$. Hence, binomial distributions with different values of $n$ can be rescaled by multiplying the function values with $\sigma$ and dividing the x-axis by $\sigma$. This is depicted in the third picture above.

The picture on the right shows shifted and normalized binomial distributions, now for more and larger values of n, in order to visualize that the function values converge to a common curve. By using Stirling's approximation of the binomial coefficients, one gets that this curve is a standard normal distribution:

$f(x)={1 \over {\sqrt {2\pi }}}\,{\rm {e}}^{-{\frac {x^{2}}{2}}}.$

This is the probability density function of the standard normal distribution ${\mathcal {N}}(0,1)$. The central limit theorem generalizes the above to limits of distributions that are not necessarily binomially distributed. The second picture on the right shows the same data, but uses a logarithmic scale, which is sometimes advisable to use in applications.

An example from sports

A soccer player makes multiple attempts to score goals. If she has a shooting success probability of 0.25 and takes 4 shots in a match, then the number of goals she scores can be modeled as B(4, 0.25). Note that p represents the probability of any given shot becoming a goal, and 1 − p represents the probability of failure. The probability of the player scoring 0, 1, 2, 3, or 4 goals on 4 shots is:

$\Pr(0{\text{ goals}})=f(0)=\Pr(K=0)={4 \choose 0}0.25^{0}(1-0.25)^{4-0}\approx 0.32$

$\Pr(1{\text{ goal}})=f(1)=\Pr(K=1)={4 \choose 1}0.25^{1}(1-0.25)^{4-1}\approx 0.42$

$\Pr(2{\text{ goals}})=f(2)=\Pr(K=2)={4 \choose 2}0.25^{2}(1-0.25)^{4-2}\approx 0.21$

$\Pr(3{\text{ goals}})=f(3)=\Pr(K=3)={4 \choose 3}0.25^{3}(1-0.25)^{4-3}\approx 0.05$

$\Pr(4{\text{ goals}})=f(4)=\Pr(K=4)={4 \choose 4}0.25^{4}(1-0.25)^{4-4}\approx 0.004$

Usage

The binomial distribution is heavily used in multiple fields, and arises naturally whenever an experiment or process can be described as a series of repeated Bernoulli trials. Examples are how many times a coin comes up "heads" if flipped repeatedly, or how many babies are born as boys in a series of births.

A limitation of the binomial distribution is that it assumes that all of the underlying Bernoulli trials have the same probability of success. The result is that the variance cannot be described independently of the probability of success and the number of trials; in particular, the variance of the observed proportion of successful outcomes is ${\frac {p(1-p)}{n}},$ decreasingly linearly with the number of trials. For example, if we observe from long-run statistics that a given congressional district with n=600,000 people votes Democratic on average 60% of the time, then if we model the votes with a binomial distribution, we predict that the standard deviation of the proportion of Democratic votes in a given election where 50% of the population votes is ${\sqrt {\frac {.6\times .4}{300,000}}}$ = 0.00089. In other words, well over 99% of the time in all such elections regardless of the candidates we expect to see between 59.7% and 60.3% Democratic votes — whereas in reality we expect to see considerably more variation. The problem here is the assumption that each individual voter votes Democratic with the same frequency as the population as a whole.

A more realistic assumption is such cases would be to place a prior distribution over the probability of success of a given Bernoulli trial (in this case, a given vote). If we make the mathematically convenient choice of using the beta distribution (the conjugate prior of the binomial distribution), then the distribution of the number of successes follows a beta-binomial distribution. This distribution is somewhat like a binomial distribution but has greater variance, and critically has an extra parameter that allows the variance to be adjusted independently of the population size and expected probability of success. As a result, a model that uses this distribution may be more realistic in such cases.

See also

• Bean machine / Galton box
• Binomial proportion confidence interval
• Logistic regression
• Multinomial distribution
• Negative binomial distribution
• Sample size