Binomial Function

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:

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) = 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.

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

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