---
title: "binomial distribution"
type: concept
related: [Normal Distribution, Probability Mass Function, Variance, Probability Of K In X Time, Probability]
source: https://www.jemoka.com/posts/kbhbinomial_distribution/
confidence: high
status: active
---

A binomial distribution is a typo of distribution whose contents are:
Binary Independent Fixed number Same probability: &ldquo;That means: WITH REPLACEMENT&rdquo; Think: &ldquo;what&rsquo;s the probability of \(n\) coin flips getting \(k\) heads given the head&rsquo;s probability is \(p\)&rdquo;.
constituents 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.
requirements 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 expected value: \(np\) variance: \(np(1-p)\) 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}
where,
\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 normal distribution approximation: \(n &gt; 20\), variance large \((np(1-p)) &gt; 10\), absolute independence; beware of continuity correction poisson distribution approximation: \(n &gt; 20\), p small \(p &lt; 0.05\) 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}
Then:
\begin{equation} X+Y \sim Bin(a+b, p) \end{equation}
