Binomial Distribution#

Definition: Binomial Distribution

We say that a [[Random Variables#Discrete Random Variable|discrete random variable]] \(X\) has a binomial distribution if there exist some \(n \in \mathbb{N}\) and some [[The Real Numbers|real number]] \(p \in [0;1]\) such that

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

for all \(k \in \{0,1,2,\dotsc, n\}\).

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

\[ X \sim \mathop{\operatorname{Bin}}(n, p) \]

By far the most common random variable which has a binomial distributions is the following. Consider some [[Experiments|experiment]] \(E_1\) with a [[Random Variables|random variable]] \(Y\) which is distributed according to a [[Bernoulli Distribution]] with parameter \(p\). Now consider the [[Experiments|experiment]] \(E_2\) which consists of repeating \(n\) times the \(E_1\). The [[Random Variables|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 [[Random Variables#Cumulative Distribution Function|cumulative distribution function]] of a [[Random Variables|discrete random variable]] \(X\) which follows the [[Binomial Distribution]] \(\operatorname{Bin}(n,p)\) is given by

\[ F(x) = \sum_{k = 0}^{\lfloor x \rfloor} \binom{n}{k} p^k (1-p)^{n-k} \]

Proof

TODO

Theorem: Probability Mass Function of Binomial Distributions

The [[Random Variables#Probability Mass Functions|probability mass function]] \(p\) of a [[Random Variables#Discrete Random Variables|discrete random variable]] \(X\) which follows the [[Binomial Distribution]] \(\mathop{\operatorname{Bin}}(n, p)\) is

\[ p(x) = \begin{cases}\binom{n}{k} p^x (1-p)^{n-x} \qquad \text{if } x \in \{0,1,2,\dotsc,n\} \\ 0 \qquad \text{otherwise}\end{cases} \]

Proof

TODO

Theorem: Mode of Binomial Distributions

The [[Random Variables#Probability Mass Functions|mode(s)]] of a [[Random Variables|discrete random variable]] which follows the [[Binomial Distribution]] \(\operatorname{Bin}(n, p)\) 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 \((n+1)p \in \{1, 2, \dotsc, n\}\);
\(n\) if \((n+1)p = n+1\).

Proof

TODO

Theorem: Expectation of Binomial Distributions

The [[Expectation]] of every [[Random Variables|random variable]] \(X\) which follows the [[Binomial Distribution]] \(\mathop{\operatorname{Bin}}(n,p)\) is given by the product of \(n\) and \(p\):

\[ \mathbb{E}(X) = np \]

Proof

TODO

Theorem: Variance of Binomial Distributions

The [[Variance and Standard Deviation|variance]] of every [[Random Variables|random variable]] \(X\) which follows the [[Binomial Distribution]] \(\mathop{\operatorname{Bin}}(n,p)\) is equal to \(np(1-p)\):

\[ \operatorname{Var}(X) = np(1-p) \]

Proof

TODO