Mathematics and Statistics · Ch 16 — Probability Distributions

Binomial Distribution

Many experiments consist of repeating the same trial a fixed number of times, where each trial has only two possible results — conventionally "success" and "failure". The count of successes then follows the binomial distribution, the most important discrete distribution in the syllabus.

Note

Bernoulli trials

A sequence of trials is a set of Bernoulli trials if: (i) there is a fixed number $n$ of trials; (ii) each trial has exactly two outcomes, success (prob. $p$ ) or failure (prob. $q = 1-p$ ); (iii) $p$ is the same in every trial; and (iv) the trials are independent.

Let $X$ = number of successes in $n$ such trials. Then $X$ can be $0, 1, 2, \ldots, n$ , and it is said to follow a binomial distribution, written $X \sim B(n, p)$ .

Note

Binomial probability formula

$P(X = r) = \binom{n}{r} p^{r} q^{\,n-r}, \qquad r = 0, 1, 2, \ldots, n, \quad q = 1-p.$

Here $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$ counts the number of ways of choosing which $r$ of the $n$ trials are the successes.

The formula splits into three honest pieces: $\binom{n}{r}$ (how many arrangements give exactly $r$ successes), $p^{r}$ (the $r$ successes), and $q^{\,n-r}$ (the remaining $n-r$ failures). Because the terms $\binom{n}{r}p^r q^{n-r}$ are exactly the terms of the expansion of $(q + p)^n$ , the probabilities add to $(q+p)^n = 1^n = 1$ , as they must.

Note

Mean and variance of the binomial distribution

$\text{Mean} = E(X) = np, \qquad \operatorname{Var}(X) = npq, \qquad \sigma = \sqrt{npq}.$ …

Definition 10Bernoulli trials

A fixed number $n$ of independent trials, each with two outcomes (success prob. $p$ , failure $q=1-p$ ) and the …

Definition 11Binomial distribution $B(n,p)$

Distribution of the number of successes $X$ in $n$ Bernoulli trials: $P(X=r)=\binom{n}{r}p^r q^{n-r}$ , with mean $np …

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