Binomial Distribution

Definition: Binomial Distribution

We say that a discrete random variable $X$ has a binomial distribution if there exist some $n\in \mathbb{N}$ and some real number $p\in [0;1]$ such that

$$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$

for all $k\in \{0,1,2,\dots ,n\}$.

NOTE

We often call $n$ and $p$ the parameters of the binomial distribution and say that $X$ is distributed according to the binomial distribution with parameters $n$ and $p$.

NOTATION

$$X\sim \mathrm{Bin}(n,p)$$

By far the most common random variable which has a binomial distributions is the following. Consider some experiment $E_1$ with a random variable $Y$ which is distributed according to a Bernoulli Distribution with parameter $p$. Now consider the experiment $E_2$ which consists of repeating $n$ times the $E_1$. The random variable $X$ which denotes the total number of times in which $Y$ was “success” follows the Binomial Distribution with parameters $n$ and $p$.

Properties

Theorem: Cumulative Distribution Function of Binomial Distributions

The cumulative distribution function of a discrete random variable $X$ which follows the Binomial Distribution $\mathrm{Bin}(n,p)$ is given by

$$F(x)=\sum _{k=0}^{\lfloor x\rfloor }\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$

PROOF

TODO

Theorem: Probability Mass Function of Binomial Distributions

The probability mass function $p$ of a discrete random variable $X$ which follows the Binomial Distribution $\mathrm{Bin}(n,p)$ is

$$p(x)=\{\begin{array}{l}\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^x(1-p{)}^{n-x}\text{if }x\in \{0,1,2,\dots ,n\}\\ 0\text{otherwise}\end{array}$$

PROOF

TODO

Theorem: Mode of Binomial Distributions

The mode(s) of a discrete random variable which follows the Binomial Distribution $\mathrm{Bin}(n,p)$ is (are):

• $\lfloor (n+1)p\rfloor$ if $(n+1)p$ is $0$ or a noninteger;

• $(n+1)p$ and $(n+1)p-1$ if $(n+1)p\in \{1,2,\dots ,n\}$;

• $n$ if $(n+1)p=n+1$.

PROOF

TODO

Theorem: Expectation of Binomial Distributions

The Expectation of every random variable $X$ which follows the Binomial Distribution $\mathrm{Bin}(n,p)$ is given by the product of $n$ and $p$:

$$\mathbb{E}(X)=np$$

PROOF

TODO

Theorem: Variance of Binomial Distributions

The variance of every random variable $X$ which follows the Binomial Distribution $\mathrm{Bin}(n,p)$ is equal to $np(1-p)$:

$$\mathrm{Var}(X)=np(1-p)$$

PROOF

TODO