Binomial distribution

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From Wikipedia, the free encyclopedia

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

Binomial distribution

Probability mass function
Cumulative distribution function
Notation	${\displaystyle �(�,�)}$
Parameters	${\displaystyle �\in \{0,1,2,\dots \}}$ – number of trials ${\displaystyle �\in [0,1]}$ – success probability for each trial ${\displaystyle �=1-�}$
Support	${\displaystyle �\in \{0,1,\dots ,�\}}$ – number of successes
PMF	${\displaystyle (\genfrac{}{}{0ex}{}{�}{�}){�}^{�}{�}^{�-�}}$
CDF	${\displaystyle {�}_{�}(�-�,1+�)}$ (the regularized incomplete beta function)
Mean	${\displaystyle ��}$
Median	${\displaystyle \lfloor ��\rfloor }$ or ${\displaystyle \lceil ��\rceil }$
Mode	${\displaystyle \lfloor (�+1)�\rfloor }$ or ${\displaystyle \lceil (�+1)�\rceil -1}$
Variance	${\displaystyle ���}$
Skewness	${\displaystyle \frac{�-�}{\sqrt{���}}}$
Ex. kurtosis	${\displaystyle \frac{1-6��}{���}}$
Entropy	${\displaystyle \frac{1}{2}{\log }_2(2�����)+�\left(\frac{1}{�}\right)}$ in shannons. For nats, use the natural log in the log.
MGF	${\displaystyle (�+�{�}^{�}{)}^{�}}$
CF	${\displaystyle (�+�{�}^{��}{)}^{�}}$
PGF	${\displaystyle �(�)=[�+��{]}^{�}}$
Fisher information	${\displaystyle {�}_{�}(�)=\frac{�}{��}}$ (for fixed ${\displaystyle �}$)

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v t e

Binomial distribution for ${\displaystyle �=0.5}$
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 ${\displaystyle 70/256}$.

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability ${\displaystyle �=1-�}$). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.^[1]

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 remains a good approximation, and is widely used.

Definitions[edit]

Probability mass function[edit]

In general, if the random variable X follows the binomial distribution with parameters n ∈ ${\displaystyle \mathbb{�}}$ and p ∈ [0,1], we write X ~ B(n, p). The probability of getting exactly k successes in n independent Bernoulli trials is given by the probability mass function:

${\displaystyle �(�,�,�)=Pr(�;�,�)=Pr(�=�)=(\genfrac{}{}{0ex}{}{�}{�}){�}^{�}(1-�{)}^{�-�}}$

for k = 0, 1, 2, ..., n, where

${\displaystyle (\genfrac{}{}{0ex}{}{�}{�})=\frac{�!}{�!(�-�)!}}$

is the binomial coefficient, hence the name of the distribution. The formula can be understood as follows: k successes occur with probability p^k and n − k failures occur with probability ${\displaystyle (1-�{)}^{�-�}}$. However, the k successes can occur anywhere among the n trials, and there are ${\displaystyle (\genfrac{}{}{0ex}{}{�}{�})}$ 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

${\displaystyle �(�,�,�)=�(�-�,�,1-�).}$

Looking at the expression f(k, n, p) as a function of k, there is a k value that maximizes it. This k value can be found by calculating

${\displaystyle \frac{�(�+1,�,�)}{�(�,�,�)}=\frac{(�-�)�}{(�+1)(1-�)}}$

and comparing it to 1. There is always an integer M that satisfies^[2]

f(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 f is maximal: (n + 1)p and (n + 1)p − 1. M is the most probable outcome (that is, the most likely, although this can still be unlikely overall) of the Bernoulli trials and is called the mode.

Example[edit]

Suppose a biased coin comes up heads with probability 0.3 when tossed. The probability of seeing exactly 4 heads in 6 tosses is

${\displaystyle �(4,6,0.3)=(\genfrac{}{}{0ex}{}{6}{4}){0.3}^4(1-0.3{)}^{6-4}=\mathrm{0.059535.}}$

Cumulative distribution function[edit]

The cumulative distribution function can be expressed as:

${\displaystyle �(�;�,�)=Pr(�\le �)=\sum _{�=0}^{\lfloor �\rfloor }(\genfrac{}{}{0ex}{}{�}{�}){�}^{�}(1-�{)}^{�-�},}$

where ${\displaystyle \lfloor �\rfloor }$ 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:^[3]

which is equivalent to the cumulative distribution function of the F-distribution:^[4]

${\displaystyle �(�;�,�)={�}_{�\text{-distribution}}\left(�=\frac{1-�}{�}\frac{�+1}{�-�};{�}_1=2(�-�),{�}_2=2(�+1)\right).}$

Some closed-form bounds for the cumulative distribution function are given below.

Properties[edit]

Expected value and variance[edit]

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:^[5]

${\displaystyle \mathrm{E}[�]=��.}$

This follows from the linearity of the expected value along with the fact that X is the sum of n identical Bernoulli random variables, each with expected value p. In other words, if ${\displaystyle {�}_1,\dots ,{�}_{�}}$ are identical (and independent) Bernoulli random variables with parameter p, then ${\displaystyle �={�}_1+\cdots +{�}_{�}}$ and

${\displaystyle \mathrm{E}[�]=\mathrm{E}[{�}_1+\cdots +{�}_{�}]=\mathrm{E}[{�}_1]+\cdots +\mathrm{E}[{�}_{�}]=�+\cdots +�=��.}$

The variance is:

${\displaystyle \mathrm{Var}(�)=���=��(1-�).}$

This similarly follows from the fact that the variance of a sum of independent random variables is the sum of the variances.

Higher moments[edit]

The first 6 central moments, defined as ${\displaystyle {�}_{�}=\mathrm{E}\left[(�-\mathrm{E}[�]{)}^{�}\right]}$, are given by

The non-central moments satisfy

and in general ^[6] [7]

${\displaystyle \mathrm{E}[{�}^{�}]=\sum _{�=0}^{�}\left\{\genfrac{}{}{0ex}{}{�}{�}\right\}{�}^{\underset{\_}{�}}{�}^{�},}$

where ${\displaystyle \left\{\genfrac{}{}{0ex}{}{�}{�}\right\}}$ are the Stirling numbers of the second kind, and ${\displaystyle {�}^{\underset{\_}{�}}=�(�-1)\cdots (�-�+1)}$ is the ${\displaystyle �}$th falling power of ${\displaystyle �}$. A simple bound ^[8] follows by bounding the Binomial moments via the higher Poisson moments:

${\displaystyle \mathrm{E}[{�}^{�}]\le {\left(\frac{�}{\log (�/(��)+1)}\right)}^{�}\le (��{)}^{�}\exp \left(\frac{{�}^2}{2��}\right).}$

This shows that if ${\displaystyle �=�(\sqrt{��})}$, then ${\displaystyle \mathrm{E}[{�}^{�}]}$ is at most a constant factor away from ${\displaystyle \mathrm{E}[�{]}^{�}}$

Mode[edit]

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

Proof: Let

${\displaystyle �(�)=(\genfrac{}{}{0ex}{}{�}{�}){�}^{�}{�}^{�-�}.}$

For ${\displaystyle �=0}$ only ${\displaystyle �(0)}$ has a nonzero value with ${\displaystyle �(0)=1}$. For ${\displaystyle �=1}$ we find ${\displaystyle �(�)=1}$ and ${\displaystyle �(�)=0}$ for ${\displaystyle �\ne �}$. This proves that the mode is 0 for ${\displaystyle �=0}$ and ${\displaystyle �}$ for ${\displaystyle �=1}$.

Let . We find

${\displaystyle \frac{�(�+1)}{�(�)}=\frac{(�-�)�}{(�+1)(1-�)}}$.

From this follows

So when ${\displaystyle (�+1)�-1}$ is an integer, then ${\displaystyle (�+1)�-1}$ and ${\displaystyle (�+1)�}$ is a mode. In the case that ${\displaystyle (�+1)�-1\notin \mathbb{�}}$, then only ${\displaystyle \lfloor (�+1)�-1\rfloor +1=\lfloor (�+1)�\rfloor }$ is a mode.^[9]