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WAMath MethodsSyllabus dot point

How do we model a single trial with exactly two outcomes, and what are its mean and variance?

Use the Bernoulli distribution to model a single two-outcome trial and find its mean and variance

WACE Year 12 Mathematics Methods Unit 3 the Bernoulli distribution: a single success-or-failure trial, its probability function, mean p and variance p times one minus p, and its role as the building block of the binomial, with a worked example.

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  1. What this dot point is asking
  2. The Bernoulli model
  3. Mean and variance
  4. Link to the binomial distribution

What this dot point is asking

SCSA Unit 3 introduces the Bernoulli distribution as the simplest discrete model: one trial, two outcomes. This dot point asks you to state its distribution and derive its mean and variance, which then generalise directly to the binomial distribution. It supports questions in both examination sections.

The Bernoulli model

A Bernoulli trial has exactly two outcomes, conventionally labelled success and failure. The random variable XX records 11 for success and 00 for failure.

Examples include a single coin toss (success = heads), one inspected item (success = defective), or one penalty kick (success = goal).

Mean and variance

Because XX takes only the values 00 and 11, the mean and variance follow directly from the definitions of expected value and variance.

The mean is

E(X)=0β‹…(1βˆ’p)+1β‹…p=p.E(X)=0\cdot(1-p)+1\cdot p = p.

For the variance, note that X2=XX^{2}=X since 02=00^{2}=0 and 12=11^{2}=1, so E(X2)=E(X)=pE(X^{2})=E(X)=p. Then

Var(X)=E(X2)βˆ’[E(X)]2=pβˆ’p2=p(1βˆ’p).\mathrm{Var}(X)=E(X^{2})-[E(X)]^{2}=p-p^{2}=p(1-p).

A binomial random variable counts the number of successes in nn independent Bernoulli trials, each with the same pp. Adding nn independent Bernoulli variables gives the binomial, which is why the binomial mean npnp and variance np(1βˆ’p)np(1-p) are simply nn times the Bernoulli values. Understanding the single-trial case makes the binomial formulae intuitive rather than memorised.