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Binomial Function

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Types of Function >

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

  1. A general type of function with two terms, used in calculus and algebra,
  2. A specific type of function, sometimes defined in terms of a power series,
  3. The binomial distribution function, used in probability,
  4. 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) = 3x2 + 2x.

2. The Binomial Function

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

fm(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 = eα 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:

Pq(n) = qn(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.


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:
Taubes, C. (2010). Lecture notes on probability, statistics and linear algebra. Retrieved December 19, 2019 from:

Stephanie Glen. "Binomial Function" From Calculus for the rest of us!

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