Continuous random variables, taught from zero

1 · From bars to curves §4.1

Module 2's variables counted things — the values sat at 0, 1, 2, … with a probability bar on each. But measure something — a resistor, a wait, a lifetime — and the possible values fill a whole interval. Between 1.2 and 1.3 there's 1.25; between 1.25 and 1.26 there's 1.255. No listing, so no bars. What replaces the pmf?

You've already met the answer, in the descriptive-statistics lesson: the density histogram, where area = proportion. Let the sample grow and the classes shrink, and the staircase of bars smooths into a curve — the probability density function (pdf) f(x). Probability is now the area under the curve, and area is what integrals compute:

A legitimate pdf needs f(x) ≥ 0 and total area ∫_−∞^∞ f(x) dx = 1 — the same two laws every pmf obeyed, with Σ traded for ∫. That trade is the whole chapter: every Module 2 formula survives with the sum replaced by an integral.

Two consequences take a moment to accept:

First: P(X = c) = 0 for every single value c. A single point is an interval of width zero — zero area. Yet some value always occurs! The resolution: probability zero doesn't mean impossible for continuous rvs; it means "a target infinitely too thin to bet on". Ask about intervals, and everything is sensible again. A pleasant side-effect: ≤ versus < no longer matters — P(X ≤ b) = P(X < b) — so Module 2's off-by-one anxiety is retired in this chapter (a genuinely relaxing fact).

Second: f(x) is not a probability. It's a probability density — probability per unit length, exactly like the descriptive-statistics lesson's histogram densities. It can exceed 1 (a uniform rv on [0, ½] has f = 2); only areas are probabilities.

The simplest pdf, the uniform distribution on [A, B], spreads probability evenly: f(x) = 1/(B−A) on the interval, 0 outside. Probabilities are just lengths ÷ total length — no integral tables needed.

2 · cdf, percentiles, mean and variance §4.2

The cdf keeps its Module 2 meaning — F(x) = P(X ≤ x) — but the running total is now a running integral: the area accumulated up to x. Instead of a staircase, a smooth S-shaped climb from 0 to 1. And the two directions of travel between f and F are the two halves of the Fundamental Theorem of Calculus (MATH U101's centrepiece, moonlighting here):

Note the last formula: F(a), not F(a−1) — single points carry no probability now. Once you have F, most questions are two lookups, exactly as in Module 2 but cleaner.

Run the machine once on a concrete pdf, f(x) = 2x on [0, 1] (a ramp — larger values more likely):

1. Valid? ∫₀¹ 2x dx = x²]₀¹ = 1 ✓.
2. cdf: F(x) = ∫₀^x 2y dy = x² for 0 ≤ x ≤ 1 (0 before, 1 after). So e.g. P(X ≤ ½) = ¼ — only a quarter of the probability sits in the left half; the ramp leans right.
3. Percentiles come from solving F(η) = p. The median is the 50th: η² = 0.5 → η = √0.5 ≈ 0.707.
4. Mean: E(X) = ∫₀¹ x · 2x dx = 2/3 ≈ 0.667 — the balance point of the ramp, and less than… wait, more? No: 2/3 < 0.707. Mean below median — the ramp's long thin tail is on the left, and the tail pulls the mean, exactly as the descriptive-statistics lesson taught.
5. Variance by the shortcut: E(X²) = ∫₀¹ x² · 2x dx = ½, so V(X) = ½ − (2/3)² = 1/18 ≈ 0.056, σ ≈ 0.236.

That five-step run — validate, integrate to F, solve for percentiles, integrate for μ, shortcut for σ² — is the §4.1–4.2 exam question, in every variation. The notes page's Worked Example 1 runs it again on a fresh pdf with every antiderivative shown.

3 · The normal distribution §4.3

Now the most important distribution in statistics — you watched it being born in Module 2's widget, where binomial bars smoothed into a bell as n grew. Measurement errors, dimensions from a stable process, heights, exam totals — anything that is the sum of many small independent effects comes out bell-shaped (Chapter 5 proves this; for now it's why the bell is everywhere). The pdf is

with two parameters you already own: μ centres the bell, σ sets its width. Here's the practical problem: that pdf has no elementary antiderivative — no formula for its cdf exists to write down. The way out is one brilliant observation: every normal is the same curve, just shifted and stretched. So tabulate one of them — the standard normal Z (μ = 0, σ = 1, cdf written Φ(z)) — and convert every other normal to it:

That number (the z-score) is the universal currency of this chapter. P(X ≤ x) = Φ((x−μ)/σ), and Φ lives in the book's Appendix table. Feel what the table contains — slide z and watch the shaded area:

The Appendix z-table is this slider, printed: one row per z, the shaded area as the entry. Three habits make it error-proof: the table gives the area to the left (right tails need 1 − Φ(z)); negative z works by symmetry (Φ(−z) = 1 − Φ(z)); and a between-two-values probability is two lookups subtracted. Anchor values worth memorising — the 68–95–99.7 rule: about 68% of any normal lies within 1σ of μ, 95% within 2σ, 99.7% within 3σ. They make superb sanity checks on every answer.

Two more standard moves, then you own §4.3:

Percentiles run the table backwards. "Find the 99th percentile" → find the z whose table entry is 0.99 (that's z = 2.33) → un-standardise: x = μ + 2.33σ. (Notation you'll meet: z_α is the z with area α to its right — so z_0.05 = 1.645.)

The normal can stand in for the binomial. When both np ≥ 10 and n(1−p) ≥ 10, Bin(n, p) ≈ N(μ = np, σ = √(npq)) — with a continuity correction: the discrete value 55 owns the strip [54.5, 55.5], so P(X ≤ 55) ≈ Φ((55.5 − μ)/σ). Try it on Bin(100, 0.5): Φ(5.5/5) = Φ(1.1) = 0.8643 — the exact answer is 0.8644. Half a unit of care, four correct decimals.

4 · Waiting times: exponential (and gamma) §4.4

Module 2 ended with the Poisson process: events striking at rate λ per unit time, counts Poisson-distributed. Natural follow-up: how long do you wait for the next event? Wait more than t ⟺ zero events in a window of length t, and Module 2 already knows that probability: e^−λt. So the waiting time X has cdf F(t) = 1 − e^−λt, and differentiating (FTC again) gives the exponential distribution:

A closed-form cdf — no tables! — which is why exponential questions are the friendliest in the chapter. The mean is instinct again: events every 1/λ time units on average. Keep the units of λ and of x consistent (λ per hour with x in hours) and half the section's traps vanish.

The exponential's famous personality trait is memorylessness: given the component has already survived time s, the chance it survives t more is exactly P(X > t) — as good as new, always. (One line shows it: P(X > s+t)/P(X > s) = e^−λ(s+t)/e^−λs = e^−λt.) It's a favourite conceptual exam question, and it's also the model's honest limitation: real machines do wear out, and that's exactly why the section introduces the gamma distribution — a two-parameter family (shape α, scale β; mean αβ, variance αβ²) built on the gamma function Γ(α), which generalises factorials (Γ(n) = (n−1)! for whole numbers). Gamma waits for the α-th Poisson event instead of the first (α = 1 recovers the exponential exactly) — Module 2's negative binomial idea, gone continuous. At this level: know the shape of the family, the mean and variance, and how to use a given Γ value; heavy gamma integrals are not first-course territory.

5 · The rest of the chapter — now officially off-syllabus §4.5–4.6

• Weibull §4.5: the engineer's lifetime workhorse — an exponential whose failure rate is allowed to change with age (a shape parameter tunes wear-out vs. infant-mortality). Reliability courses live on it.
• Lognormal §4.5: X is lognormal when ln(X) is normal — the model for quantities built by multiplying many small effects (incomes, particle sizes, repair times). Computations reduce to §4.3 with a log first.
• Beta §4.5: lives on a bounded interval [A, B] — the tool for proportions and task-completion fractions.
• Probability plots §4.6: the practical question "is my data plausibly normal?" answered by plotting sorted data against theoretical percentiles — a straight-ish line says yes. Concept in one sentence; the skill is reading the plot, and it pairs beautifully with the R-lab page's boxplots.

6 · You're ready — then on to Part 2

The chapter in one sentence: replace Module 2's sums with integrals, and the same framework — pdf, cdf, percentiles, E and V — runs on measured quantities, with the normal (standardise, then one table) and the exponential (closed-form cdf, memoryless) as the two stars. The handout adds smaller topics to this module: chi-squared is covered above, and MGFs for continuous rvs, Chebyshev's inequality (stated, not proved) and transformation methods now have their own page, the Module 3 Part 2 lesson (with notes). The lectured core above is the bulk of the module. Next:

• Work the examples: the notes page has the discrete↔continuous dictionary, three fully computed examples (pdf workout, normal resistors, exponential lifetimes), and the practice table.
• Revise from the notes page, not this one. The course guide now carries the handout's confirmed scope and the evaluation calendar.