Introduction to Statistical Modelling Revision

Setting priorities

Modules 1–3 each have 15 hours. Module 4 has 30 hours and explicitly requires a practical record with at least 10 problems. The outcomes emphasise understanding random-variable/distribution theory, using distributions, analysing applications, and evaluating fitting/simulation with spreadsheets. The assessment lists 2-, 6- and 12-mark theory answers plus a practical evaluation. The priority labels below follow that evidence and do not predict marks for any one topic.

MUST MASTER

PMF/PDF/CDF, expectation and variance, standard distribution calculations, normal table use, and spreadsheet problem solving.

IMPORTANT

Transformations, joint and conditional distributions, generating functions, memorylessness and model selection.

QUICK-SCORE

Definitions, supports, parameter meanings, standard properties and distribution-identification clues.

Scope reminder

Teacher-specific content is listed without topics in the supplied syllabus. Do not assume additional topics for that module without lecturer guidance.

MODULE 1 · 15 HOURS

Random variable theory

10 core concepts

Random variable
Discrete and continuous cases
PMF
PDF
CDF
Univariate transformation
Bivariate random variable
Joint distribution
Marginal and conditional distributions
Independence of random variables

Key terms

Support · mass · density · cumulative · transformation · joint · marginal · conditional · independent · distribution function

Essential formula prompts

Check total PMF sum or PDF integral is 1; F(x)=P(X≤x); derive marginals by summing/integrating; conditional = joint divided by marginal; independence means joint factors into marginals.

Common mistakes: treating PDF height as point probability; leaving a transformed variable’s support unstated; confusing joint and marginal; treating one matching cell in an independence check as proof of independence.

Expected / Exam-Focused Questions based on the syllabus

2 marks: Define a discrete random variable; state a property of a CDF; define marginal distribution.
6 marks: Explain PMF, PDF and CDF and their properties; obtain marginal/conditional probabilities from a joint table.
12 marks: Given a bivariate distribution, find marginal and conditional distributions and assess independence, showing checks.
Numerical: derive a simple CDF from a PMF; transform a discrete variable; calculate marginal and conditional probabilities.

Checklist

I can distinguish discrete from continuous random variables.
I can validate a PMF/PDF and calculate a CDF.
I can find the support and distribution of a simple transformation.
I can obtain joint, marginal and conditional distributions.
I can test independence using all relevant values.

MODULE 2 · 15 HOURS

Mathematical expectation

10 core concepts

Probability-weighted average
Expectation of a discrete variable
Expectation of a continuous variable
Linearity
Expectation of a function
Variance
Mean deviation
AM, GM and HM using expectation
Moment-generating function
Characteristic function

Key terms and formulas

Expected value · raw moment · central moment · variance · standard deviation · MGF · characteristic function · derivative · parameter

E(X)=Σxp(x) or ∫xf(x)dx; Var(X)=E(X²)−[E(X)]²; M(t)=E(eᵗˣ); φ(t)=E(eⁱᵗˣ).

Common mistakes: confusing E(X²) with [E(X)]²; forgetting the square on scaling in Var(aX+b); claiming variance of a sum always adds; mixing up an MGF with a characteristic function.

Expected / Exam-Focused Questions based on the syllabus

2 marks: Define mathematical expectation; state the variance identity; define MGF.
6 marks: Explain expectation and its properties with a simple example; calculate mean and variance from a distribution.
12 marks: Explain expectation, variance and generating functions, then solve a supplied simple distribution problem.
Numerical: compute E[g(X)], mean deviation or variance; differentiate a simple MGF to obtain moments.

Checklist

I can calculate expectation for discrete and continuous examples.
I can apply linearity and variance properties with their conditions.
I can express mean and variance using expectations.
I can describe the syllabus-level role of MGF and characteristic functions.
I can show each substitution and interpret the result.

MODULE 3 · 15 HOURS

Discrete and continuous distributions

10 core concepts

Discrete uniform
Bernoulli
Binomial
Poisson
Geometric
Geometric lack of memory
Continuous uniform
Exponential
Gamma
Normal and standard normal

Key terms

Support · parameter · success · rate · shape · density · tail probability · standardisation · z-score · memorylessness

Write the parameter convention: e.g. gamma shape-rate or shape-scale; geometric trials-to-first-success or failures-before-success.

Distribution identification: fixed number of same-chance trials → binomial; count in an interval → Poisson; trials to first success → geometric; continuous wait → exponential; bell-shaped symmetric quantity → normal.

Expected / Exam-Focused Questions based on the syllabus

2 marks: Define Bernoulli distribution; state the mean of Poisson; define standard normal variable.
6 marks: Explain conditions/properties of binomial or Poisson; describe lack of memory for geometric/exponential; compare selected distributions.
12 marks: Discuss listed distributions with their supports, parameters, key properties and applications; solve an appropriate probability problem.
Numerical: binomial/Poisson/geometric probabilities; uniform/exponential tail; gamma parameters; normal probabilities using a standard table.

Checklist

For each model I know what it describes, support and parameter meanings.
I can identify whether a scenario is discrete or continuous.
I can calculate standard probabilities without mixing formulas.
I can explain the geometric/exponential lack-of-memory property.
I can standardise a normal value and read the correct table area.

MODULE 4 · 30 HOURS

Spreadsheet for statistical computing

10 key practical skills

Enter, label and validate data
Use cell references and ranges
Calculate and check a PMF/CDF
Find marginal probabilities
Calculate expectation and variance
Use a Bernoulli/binomial function
Use Poisson/geometric calculations
Calculate continuous-model probabilities
Use normal distribution/table functions
Simulate outcomes and compare empirical/theoretical results

The record must contain at least 10 problems. For each, include the objective, inputs, formula/function, output, verification and interpretation.

Common mistakes: wrong cell range, mistaken rate/scale, incorrect cumulative vs point-probability function, failing to check that probabilities sum to 1, and assuming one random simulation run must equal theoretical probability.

Checklist

I can use the spreadsheet available in my class and explain function inputs.
I can verify selected results manually or with distribution totals.
I can explain how a simulation approximates a probability.
I have at least 10 complete syllabus-related practical record problems.

Practice question bank and model answer guidance

Questions are syllabus-aligned practice. They are not actual MGU questions unless an official paper is separately supplied.

2-mark questions

Define a PMF and state its total-probability property.
What is a CDF?
Define mathematical expectation.
State the mean and variance of a Bernoulli variable.
State the lack-of-memory property of the exponential distribution.
What is a standard normal variable?

6-mark questions

Explain the difference between PMF, PDF and CDF.
From a joint PMF table, find marginal and conditional distributions and test independence.
Calculate expectation and variance from a discrete distribution, showing steps.
Compare binomial, Poisson and geometric models, giving a suitable use for each.
Explain how to standardise a normal variable and use the table for a probability.
Describe a spreadsheet method for simulating and checking a binomial probability.

12-mark questions

Explain random variables and their probability functions; include univariate and bivariate examples and properties.
Discuss expectation, variance and generating functions, with a simple worked problem.
Describe the discrete and continuous distributions in the syllabus, including parameters, properties and suitable applications.
Plan a spreadsheet analysis and simulation for a distribution problem; show checks and interpret the output.

Numerical practice

X takes 0,1,2 with probabilities 0.2,0.5,0.3. Find F(0), F(1), E(X), E(X²) and Var(X).
For the joint table in Module 1, find P(X=0), P(Y=0|X=1), and determine whether X and Y are independent.
A fair coin is tossed 5 times. Find the probability of exactly 3 heads.
For a Poisson model with λ=2, find P(X=1) and P(X=0).
For geometric p=0.25 under trials-to-first-success convention, find P(X=3) and E(X).
If X~Uniform(4,10), find P(5<X<8), E(X), and Var(X).
If X is exponential with rate 0.2, find P(X>5).
If X~N(50,8²), standardise x=62 and use the provided table value to obtain a tail probability.

How to shape model answers

2 marks: precise definition plus one key property or example.

6 marks: definition, explanation, formula with symbols, short example or property, and interpretation.

12 marks: organised introduction, concepts and conditions, formulas, stepwise worked problem or diagram, interpretation and concise conclusion.

In numerical answers show: Question → Given → Required → Formula → Substitution → Calculation → Answer → Interpretation.

7-day revision plan

Day	Focus and output
1	Random variables, support, PMF/PDF/CDF; validate examples and make a one-page comparison.
2	Transformations and bivariate/joint distributions; solve marginal and conditional table problems.
3	Expectation, variance, mean deviation and expectation forms of averages; solve raw PMF examples.
4	MGF and characteristic function definitions/properties; practise simple MGF derivatives.
5	Discrete distributions: uniform, Bernoulli, binomial, Poisson and geometric; solve one of each.
6	Continuous uniform, exponential, gamma, normal and z-table problems.
7	Spreadsheet practice, verify 10-record requirement, mixed timed questions and error review.

14-day revision plan

Day	Focus
1	Discrete vs continuous variables, support and examples.
2	PMF and PDF properties; easy probability questions.
3	CDFs, interval probabilities and distribution function properties.
4	Univariate transformations, including the transformed range.
5	Joint PMF/PDF and joint CDF concepts.
6	Marginal/conditional distributions and independence.
7	Expectation, its properties and expectation of a function.
8	AM/GM/HM, mean deviation, variance and standard deviation.
9	MGF and characteristic function definitions, properties and examples.
10	Discrete uniform, Bernoulli and binomial practice.
11	Poisson and geometric, including lack of memory.
12	Continuous uniform, exponential and gamma.
13	Normal distribution, standardisation and table practice.
14	Spreadsheet record check, mixed questions, answer writing and error review.

Final 24-hour checklist

I can define each distribution and name its support and parameters.
I can distinguish PMF, PDF and CDF and check probability totals.
I can calculate expectation and variance carefully.
I can select the right distribution for a described situation.
I can work a standard normal table question and check its tail direction.
I know my parameter conventions and common formula traps.
My practical record has the required minimum of 10 problems.

If you can do these things, you are exam ready: explain the model, identify when it applies, calculate without skipping steps, interpret the answer, check it, and communicate it in a clear examination response.

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