Discrete probability distributions, including the uniform discrete distribution and an introduction to the Bernoulli distribution, with calculations of expected value and variance

What this dot point is asking

QCAA wants Year 11 students to introduce discrete probability distributions, compute expected value and variance, and recognise the Bernoulli distribution. Foundation for Unit 3 binomial work.

Discrete random variable

$X$ takes values in a finite or countable set.

Probability mass function (pmf): $P(X = x)$ for each value $x$.

Properties: $0 \leq P(X = x) \leq 1$; $\sum P(X = x) = 1$.

Expected value

Interpretation: long-run average.

Linearity: $E(aX + b) = a E(X) + b$.

Variance and standard deviation

where $E(X^2) = \sum x^2 \cdot P(X = x)$.

Standard deviation $\sigma = \sqrt{\text{Var}(X)}$.

Property: $\text{Var}(aX + b) = a^2 \text{Var}(X)$.

The Bernoulli distribution

One trial with two outcomes: success (probability $p$), failure (probability $1 - p$).

$X = 1$ for success, $X = 0$ for failure.

$E(X) = p$. $\text{Var}(X) = p(1 - p)$.

The uniform discrete distribution

Each of $n$ outcomes is equally likely with probability $1/n$.

For values $1, 2, \ldots, n$: $E(X) = (n+1)/2$, $\text{Var}(X) = (n^2 - 1)/12$.

Worked example

A fair die: $X$ = number rolled.

$E(X) = (1+2+3+4+5+6)/6 = 21/6 = 3.5$.

$E(X^2) = (1+4+9+16+25+36)/6 = 91/6 \approx 15.17$.

$\text{Var}(X) = 91/6 - 12.25 = 35/12 \approx 2.92$.

Common errors

Variance formula error. $E(X^2) - [E(X)]^2$, not $E(X^2) - E(X)$.

Forgetting probabilities sum to 1. Always check.

Missing $\sigma = \sqrt{\text{Var}}$ step. Variance is squared; standard deviation requires the square root.

In one sentence

A discrete random variable $X$ has probability mass function $P(X=x)$ summing to 1, with expected value $E(X) = \sum x P(X=x)$ and variance $\text{Var}(X) = E(X^2) - [E(X)]^2$; the Bernoulli distribution (one trial, success probability $p$, $E(X) = p$, $\text{Var}(X) = p(1-p)$) is the foundation for Unit 3 binomial.