Binomial Distribution

Topics: Probability

Definition

The Binomial Distribution is a discrete probability distribution with parameters $n$ and $k$. It consists of a sequence of $n$ independent trials, where each trial has 2 outcomes: success, with probability $p$, and failure, with probability $1-p$.
Each trial is called a Bernoulli Trial.

Probability Mass Function

The probability of getting exactly $k$ successes in $n$ independent trials is given by the probability mass function (discrete probability distribution function):
$f(k,n,p)=Pr(X=k)=\begin{pmatrix} n \\ k \end{pmatrix}p^k(1-p)^{n-k}$
where $\begin{pmatrix} n \\ k \end{pmatrix}=\frac{n!}{k!(n-k)!}$ is the binomial coefficient.

Cumulative Distribution Function

The Cumulative Distribution Function is expressed as:
$F(k,n,p)=Pr(X\leq k)=\sum_{i=0}^{\lfloor k \rfloor} \begin{pmatrix}n \\ i \end{pmatrix} p^i (1-p)^i$
where $\lfloor k \rfloor$ is the floor under $k$, that it, the greatest integer less than or equal to $k$.

Special Cases

• The Binomial Distribution is a special case of the Multinomial Distribution
• The Bernoulli Distribution is a special case of the Binomial Distribution where $n=1$

References

1. https://en.wikipedia.org/wiki/Binomial_distribution