Binomial Distribution

Compact study note.

Summary

The binomial distribution counts successes in a fixed number of independent Bernoulli trials with common success probability.^[1]

Prerequisites

• Bernoulli Distribution
• Binomial Theorem

Definition

$X\sim \mathrm{Binomial}(n,p)$ when $X=\sum _{i=1}^n{X}_i$ for IID $X_i\sim \mathrm{Bernoulli}(p)$ .

Notation and Assumptions

Trials are independent, $n$ is fixed before observing data, and $p$ is constant across trials.

Parameters

$n\in \{0,1,2,\dots \}$ and $0\le p\le 1$ .

Support

$\{0,1,\dots ,n\}$ .

PMF or PDF

$P(X=k)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$ for $k=0,\dots ,n$ .

CDF

$F_X(k)=\sum _{j=0}^{\lfloor k\rfloor }(\genfrac{}{}{0ex}{}{n}{j})p^j(1-p{)}^{n-j}$ .

Moments

$E[X]=np$ , $\mathrm{Var}(X)=np(1-p)$ , and $M_X(t)=(1-p+p{e}^t{)}^n$ .

Essential Result

The binomial model fixes the number of trials and randomizes the number of successes.

Small Example

For five fair coin flips, $P(X=3)=(\genfrac{}{}{0ex}{}{5}{3})(0.5{)}^3(0.5{)}^2=0.3125$ .

Common Mistakes

• Using binomial sampling for draws without replacement from a small population.
• Forgetting the independence and constant- $p$ assumptions.

Connections

• Bernoulli Distribution
• Hypergeometric Distribution
• Negative Binomial Distribution

References