Binomial Distribution

probability-theory

Definition

Binomial Distribution

The Binomial distribution models the number of success in $n$ independent Bernoulli trials with success probability $p$.

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

Probability Mass Function

The probability mass function is given by:

$$f(x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x}$$

for $x\in \{0,1,..,n\}$ with $p\in [0,1]$…

Expected Value

We can write $X$ (random variable) as the sum of $B_i\stackrel{\text{iid}}{\sim }\mathrm{Bernoulli}(p)$ and use the linearity of the expectation:

$$\mathbb{E}[X]=\sum _{i=1}^n\mathbb{E}[B_i]=\sum _{i=1}^{n}p=np$$

Variance

Using that $X$ (random variable) is a sum of independent and identically distributed Bernoulli trials gives a variance:

$$\mathbb{V}ar(X)=\sum _{i=1}^n\mathbb{V}ar(B_i)=np(1-p)$$