# Binomial Probability Distribution Family

A Binomial Probability Distribution Family, [math]\displaystyle{ B(n,p) }[/math], is a finite support discrete probability distribution family for a binomial random variable.

**Context:**- It can be referenced by a Binomial Probability Function.
- It has Binomial Probability Function Mean of [math]\displaystyle{ n p }[/math].

**Example(s):**- http://itl.nist.gov/div898/handbook/eda/section3/gif/binpdf4.gif
- a Bernoulli Distribution, where [math]\displaystyle{ n=1 }[/math].
- a Beta-Binomial Probability Distribution.
- …

**Counter-Example(s):****See:**Coin-Toss Experiment, Rule of Succession.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/binomial_distribution Retrieved:2015-6-15.
- 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 yes/no experiments, each of which yields success with probability*p*.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 probability theory and statistics, the
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/binomial_distribution#Specification Retrieved:2015-6-15.
- 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: : [math]\displaystyle{ f(k;n,p) = \Pr(X = k) = {n\choose k}p^k(1-p)^{n-k} }[/math] for k*= 0, 1, 2, ...,*n*, where : [math]\displaystyle{ {n\choose k}=\frac{n!}{k!(n-k)!} }[/math] is the binomial coefficient, hence the name of the distribution. The formula can be understood as follows: we want exactly*p*k*successes (^{k}) and*n*−*k*failures (1 − p*)*n^{}*−*k*. However, the*n trials, and there are [math]\displaystyle{ {n\choose k} }[/math] different ways of distributing*k*successes can occur anywhere among the*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]\displaystyle{ f(k,n,p)=f(n-k,n,1-p). }[/math] Looking at the expression*ƒ*(*k*,*n*,*p*) as a function of*k*, there is a*M*k*value that maximizes it. This*k*value can be found by calculating : [math]\displaystyle{ \frac{f(k+1,n,p)}{f(k,n,p)}=\frac{(n-k)p}{(k+1)(1-p)} }[/math] and comparing it to 1. There is always an integer*that satisfies :[math]\displaystyle{ (n+1)p-1 \leq M \lt (n+1)p. }[/math]**(*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.** Recurrence relation [math]\displaystyle{ \left\{p (n-k) \text{Prob}(k)+(k+1) (p-1) \text{Prob}(k+1)=0,\text{Prob}(0)=( 1-p)^n\right\} }[/math]*

- In general, if the random variable

### 2005

- (Lord et al., 2005) ⇒ Dominique Lord, Simon P. Washington, and John N. Ivan. (2005). “Poisson, Poisson-gamma and zero-inflated regression models of motor vehicle crashes: balancing statistical fit and theory.” In: Accident Analysis & Prevention, 37(1). doi:10.1016/j.aap.2004.02.004
- QUOTE: The mean and variance of the binomial distribution are [math]\displaystyle{ E(Z) = Np }[/math] and [math]\displaystyle{ VAR(Z) = Np(1-p) }[/math] respectively.

### 2002

- QuickCalcs Online Calculator: http://www.graphpad.com/quickcalcs/probability1.cfm
- QUOTE: The binomial distribution applies when there are two possible outcomes. You know the probability of obtaining either outcome (traditionally called "success" and "failure") and want to know the chance of obtaining a certain number of successes in a certain number of trials.
- How many trials (or subjects) per experiment?
- What is the probability of "success" in each trial or subject?

- QUOTE: The binomial distribution applies when there are two possible outcomes. You know the probability of obtaining either outcome (traditionally called "success" and "failure") and want to know the chance of obtaining a certain number of successes in a certain number of trials.