**The term “binomial function” can mean a few different things:**

- A
**general type of function with two terms**, used in calculus and algebra, - A specific type of function, sometimes defined in terms of a
**power series**, - The
**binomial distribution function**, used in probability, - A function used in
**mathematical software**to calculate binomial probabilities.

## 1. A Binomial Function of Two Terms

“A” binomial function is a function with two terms (Dick & Patton, 1992). Examples:

- f(x) = 2x + 2
- f(x) = 3x
^{2}+ 2x.

## 2. **The** Binomial Function

“The” binomial function is a specific function with the form:

**f _{m}(x) = (1 + x)^{m}**

Where “m” is a real number. If m is positive, the function is a polynomial function.

Other forms of binomial functions are used throughout calculus. For example, as a power series expansion, the binomial function is defined for any real number α:

**(1 + t) ^{α} = e^{α log ( + t)}**

## Binomial Probability Function

In probability and statistics, The Binomial Probability Function is sometimes just called *the binomial function*.

The generic form of the binomial probability function is:

P_{q}(n) = q^{n}(1 – q)^{N-n}

Where “p” is the probability of a success and q is the probability of failure, defined for the set {0,…, N).

“Successes” and “Failures” are defined by what experiment you’re performing, not by success or failure of the entire experiment. For example, if you’re trying to find the probability of picking a red ball from a jar of red and black balls, your “success” would be pulling out a red ball and a “failure” would be pulling a black.

## Use in Mathematical Software

Binomial functions are used in software to calculate binomial probabilities. For example, in R, dbinom(x,n,p) finds the number of successes for a certain number of trials.

## References

Dick, T. & Patton, C. Calculus, Volume 1. PWS-Kent Publishing Company.

Nagy, G. Binomial functions and Taylor series (Sect. 10.10). Retrieved December 19, 2019 from: https://users.math.msu.edu/users/gnagy/teaching/12-spring/mth133/L35-133.pdf

Taubes, C. (2010). Lecture notes on probability, statistics and linear algebra. Retrieved December 19, 2019 from: http://people.math.harvard.edu/~knill/teaching/math19b_2011/handouts/chapters1-19.pdf

**CITE THIS AS:**

**Stephanie Glen**. "Binomial Function" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/binomial-function/

**Need help with a homework or test question? **With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!