HSC Distributions: Study Guide & Outline

This chapter sets the foundation by reviewing prerequisite probability concepts from the Extension 1 syllabus. It defines random experiments and random variables, distinguishes discrete and continuous types, and recalls discrete probability distributions, including the uniform discrete distribution. Bernoulli trials and binomial experiments are introduced as key building blocks. The chapter also prepares students for continuous distributions by covering improper integrals and gives a first look at the Central Limit Theorem, leading to Gaussian (normal) distributions and the empirical rule. Core formulas and ideas are summarised for quick reference.

This chapter formalises continuous random variables through relative frequency and probability density functions (PDFs). It defines the mean and variance of a continuous distribution via integration, establishing the expected value $E(X) = \int x f(x)\,dx$ and $\operatorname{Var}(X) = E(X^2) - [E(X)]^2$. The standard normal distribution is introduced with its bell-shaped PDF $\phi(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}$ and cumulative probabilities are explored using statistical tables or technology. General normal distributions $N(\mu,\sigma^2)$ are then treated through standardisation $Z = \frac{X-\mu}{\sigma}$, enabling applications to real-world data. Investigations encourage empirical exploration of normality and the 68–95–99.7 rule.

Building on Bernoulli trials, this chapter defines the binomial random variable $X \sim \operatorname{Bin}(n,p)$ and its probability mass function $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$. The mean $np$ and variance $np(1-p)$ are derived. It then investigates the shape of binomial distributions for different $n$ and $p$, leading to the normal approximation when $np$ and $n(1-p)$ are sufficiently large, justified by the Central Limit Theorem. The concept of sample proportions $\hat{p} = X/n$ is introduced, and its approximate normal distribution $\hat{p} \approx N(p, \frac{p(1-p)}{n})$ is used for inference.

This chapter extends students' understanding with more sophisticated probability models and computational techniques. Monte Carlo simulations are introduced as a method for approximating probabilities using random number generation. The hypergeometric distribution is examined as an alternative to the binomial when sampling without replacement, with its probability formula $P(X=k) = \frac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}}$. A spreadsheet lab guides students in simulating the Central Limit Theorem by repeatedly sampling from non-normal populations and observing the convergence of sample means to normality.

This section provides detailed, step-by-step solutions to a graduated set of problems. Basic problems reinforce core skills: generating and interpreting random numbers, applying linear transformations $E(aX+b)=aE(X)+b$, $\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$, and working with stepped probability density functions. Medium problems tackle transformed uniform distributions, asymmetric triangular distributions, the variance identity $\operatorname{Var}(X)=E[(X-\mu)^2]$, and real-world scenarios like clinical trials and fundraising. Advanced problems delve into the exponential distribution $f(x)=\lambda e^{-\lambda x}$, higher moments $E(X^k)$, kurtosis of the normal distribution, and statistical analysis of heart rate data, introducing moment generating functions and integration techniques for continuous models.

This chapter offers a wide range of exercises for consolidation and extension. Warm-up drills start with simple probability using spinners. Stretch problems cover dice probabilities, simulating binomial distributions with coin tosses, calculating normal probabilities with Python code, rare blue variants under sample proportions, complex inheritance scenarios, and the memoryless property of the geometric distribution $P(X>k) = (1-p)^k$. The Challenge Corner presents advanced mathematical curiosities: Benford's law for leading digits $P(d)=\log_{10}(1+1/d)$ and the squared Cauchy distribution, inviting students to explore non-standard distributions and their properties.

The final chapter summarises the key learning points of the booklet, connecting discrete and continuous distributions and highlighting the unifying role of the Central Limit Theorem. It reflects on the journey from basic probability to advanced simulations and distributions, encouraging students to appreciate the power of statistical thinking. Suggestions for further reading and university-level topics are provided.

This appendix provides a comprehensive table of the standard normal cumulative distribution function $\Phi(z) = P(Z \le z)$, essential for solving problems without technology. It includes instructions on using the table to find probabilities and $z$-scores.

Offering a bridge to tertiary statistics, this appendix introduces continuous distributions beyond the HSC syllabus. The exponential, gamma, and beta distributions are presented with their probability density functions, parameters, and typical applications (e.g., waiting times, Bayesian analysis). Mathematical Expectation and moment generating functions are briefly reviewed to unify the treatment.