Houjun Liu

binomial distribution

A binomial distribution is a typo of distribution whose contents are:

  • Binary
  • Independent
  • Fixed number
  • Same probability: “That means: WITH REPLACEMENT”

Think: “what’s the probability of \(n\) coin flips getting \(k\) heads given the head’s probability is \(p\)”.


We write:

\begin{equation} X \sim Bin(n,p) \end{equation}

where, \(n\) is the number of trials, \(p\) is the probability of success on each trial.


Here is the probability mass function:

\begin{equation} P(X=k) = {n \choose k} p^{k}(1-p)^{n-k} \end{equation}

additional information

properties of binomial distribution

deriving the expectation

The expectation of the binomial distribution is derivable from the fact:

\begin{equation} X = \sum_{i=1}^{n} Y_{i} \end{equation}


\begin{equation} \begin{cases} X \sim Bin(n,p) \\ Y_{i} \sim Bern(p) \end{cases} \end{equation}

Now, recall that expected value is linear.

Therefore, we can write that:

approximating binomial

adding binomial distribution

For \(X\) and \(Y\) independent binomial distributions, with equivalent probability:

\begin{equation} X \sim Bin(a, p), Y \sim Bin(b, p) \end{equation}


\begin{equation} X+Y \sim Bin(a+b, p) \end{equation}