The Binomial Distribution

The binomial distribution applies to repeated independent trials with only two possible outcomes each — heads/tails, defective/good, and so on — the probability of each sequence computed via the multiplication rule (or a tree diagram).

Motivating derivation. Toss a coin once: $X\sim\text{Ber}(p)$. Toss it $n$ times and let $X$ = number of heads; $X$ takes the values $0,1,\dots,n$, and getting exactly $x$ heads means choosing which $\binom nx$ of the $n$ tosses were heads, each such arrangement having probability $p^x(1-p)^{n-x}$:

The name "binomial" comes directly from the binomial expansion $(a+b)^n=\sum_{x=0}^n\binom nx a^xb^{n-x}$: taking $a=p,\ b=1-p$ shows the probabilities sum to $(p+(1-p))^n=1^n=1$, as they must — since each trial's outcome is classified into exactly two ("bi-") categories.

Definition 11.11 (binomial random variable). $X$ is a binomial random variable if (i) the $n$ repeated trials are independent, with $n$ finite; (ii) each trial has exactly two outcomes, "success" or "failure"; (iii) the success probability $p$ is constant across trials.

Definition 11.12 (binomial distribution). With $q=1-p$ the failure probability, $X\sim B(n,p)$ has pmf

Mean and variance: $\mu=E(X)=np$ and $\sigma^2=V(X)=np(1-p)=npq$. …