ECON1012 · Data Analytics
Random Variables & the Normal Distribution
Random Variables & the Normal Distribution (Module 4, Week 4) opens the Inferential Statistics block of ECON 1012 by turning Week 3's event probabilities into a general framework: a random variable assigns a number to every outcome, and its probability distribution lets you compute the likelihood of any event at once. You classify variables as discrete or continuous, check that a discrete distribution's probabilities sum to 1, and compute its expected value μ = Σx·p(x), variance and standard deviation — plus the transformation laws E(cX + b) = cE(X) + b and V(cX + b) = c²V(X). The second half introduces density functions and the normal distribution, the workhorse for the rest of the semester: standardise with Z = (X − μ)/σ and read cumulative left-tail areas from the Z table — the same table printed inside the final exam paper.
What this chapter covers
- 01Random variable: a rule assigning a number to each outcome — discrete (countable values) vs continuous (uncountable)
- 02Discrete distribution requirements: 0 ≤ p(x) ≤ 1 and Σp(x) = 1
- 03Expected value μ = E(X) = Σx·p(x); variance σ² = E(X²) − μ²; σ = √V(X)
- 04Laws of E and V: E(cX + b) = cE(X) + b · V(cX + b) = c²V(X)
- 05Continuous variables: P(X = a) = 0; probability = area under the density f(x)
- 06Normal distribution: bell-shaped, symmetric, fully defined by μ and σ²
- 07Standardisation Z = (X − μ)/σ; the course Z table gives area to the LEFT
- 08Inverse-Z: search the table body for the probability, read z, then x = μ + zσ
Normal probabilities in both directions with the Z table
- 1 mark(a) Standardise the cutoff: Z = (50 − 42)/8 = 8/8 = 1.00.
- 1 mark(a) The table gives area to the left: P(Z < 1.00) = 0.8413, so P(X > 50) = 1 − 0.8413 = 0.1587 — about 15.9% of deliveries.
- 1 mark(b) Standardise both ends: Z = (30 − 42)/8 = −1.50 and Z = 1.00 (from part a).
- 2 marks(b) P(30 < X < 50) = P(Z < 1.00) − P(Z < −1.50) = 0.8413 − 0.0668 = 0.7745, i.e. about 77.5%.
- 2 marks(c) 'Only 5% take longer than t' means P(X > t) = 0.05, so P(Z < z) = 0.95. Searching the body of the Z table for 0.95 gives z = 1.645.
- 1 mark(c) Un-standardise: t = μ + zσ = 42 + 1.645 × 8 = 42 + 13.16 = 55.16 ≈ 55.2 minutes.
Key terms
- Random variable
- A function or rule that assigns a numerical value to each outcome of a random experiment; discrete if it takes a countable number of values, continuous if its values are uncountable.
- Probability distribution
- A table, formula, or graph pairing each value of a random variable with its probability; for a discrete variable every p(x) must lie between 0 and 1 and the probabilities must sum to 1.
- Expected value E(X)
- The weighted average of a random variable's possible values, with the probabilities as weights: μ = E(X) = Σx·p(x). It carries the variable's units and is the distribution's long-run mean.
- Variance of a random variable
- The probability-weighted spread σ² = V(X) = Σ(x − μ)²p(x), usually computed via the shortcut σ² = E(X²) − μ²; the standard deviation σ = √V(X) returns the answer to native units.
- Probability density function
- The curve f(x) describing a continuous random variable: f(x) ≥ 0 everywhere and the total area under the curve equals 1. P(a ≤ X ≤ b) is the area between a and b, and P(X = a) = 0 for any single value.
- Standard normal distribution
- The normal distribution with μ = 0 and σ = 1. Any normal X converts to it via Z = (X − μ)/σ, which is why a single Z table handles every normal-probability question.
Random Variables & the Normal Distribution FAQ
How do I read the Z table in ECON 1012?
The course table is cumulative: each entry is the area to the LEFT of z, P(Z < z). Right-tail questions use the complement P(Z > z) = 1 − P(Z < z), and symmetry gives P(Z < −z) = P(Z > z). Both the Z and t tables are printed inside the final exam paper, so practise reading them by hand — the exam is a hand-calculation paper, not an Excel one.
Do I need the binomial formula in Week 4?
No. Module 4 covers generic discrete distributions — any table of values x with probabilities p(x) — plus the normal distribution. The binomial only appears in Week 5, and there through its mean np and variance np(1 − p) as a lead-in to the sample proportion, not through a probability formula. If in doubt, check the current module on myLearning.
Why is P(X = a) exactly zero for a continuous random variable?
Because probability for a continuous variable is area under the density curve, and the region above a single point has zero width. That is why continuous questions always ask about intervals, and why P(X ≤ a) and P(X < a) give the same answer for a normal variable — a convenience worth exploiting, not worrying about.
How much of the ECON 1012 final exam builds on this module?
A lot. The final (25 MCQs + 3 case-study questions, 180 minutes, covering Weeks 1-10) can test it directly — classifying discrete vs continuous variables, computing E(X) and V(X) from a table, normal tail and inverse-normal questions — and it feeds everything after: Z = (X − μ)/σ is the engine behind sampling distributions, confidence intervals and hypothesis tests in Weeks 5-7. Get fluent with the table now and the inference weeks get much lighter.
Studying with AI? Sia — free AI economics tutor works through ECON 1012 step by step.
Exam move
Drill the two halves separately. For discrete distributions, use the shortcut σ² = E(X²) − μ² and carry units on E(X) and σ — practice-exam-style MCQs deliberately pair a correct mean with a wrong variance. For transformations, variance picks up c², not c: if Y = cX + b, then sd(Y) = |c|·sd(X) and the + b changes nothing. For normal problems, sketch and shade before touching the table — the course table is strictly area-to-the-left, so convert with P(Z > z) = 1 − P(Z < z) and P(Z < −z) = P(Z > z). For inverse questions, search the table body for the probability, read z from the margins, then un-standardise with x = μ + zσ. Put these identities on your A4 note sheet early.
