MAST90105 · Methods Of Mathematical Statistics
Continuous Random Variables and the CLT
A continuous random variable carries its probability in a density f(x) rather than a point-mass PMF: probabilities are areas, P(a ≤ X ≤ b) = ∫abf, and any single point has probability zero. The chapter teaches the working moves — find the normalising constant that makes the density integrate to one, build the CDF by integrating, then read off the mean, median and variance — and introduces the continuous families: the exponential (memoryless waiting times), the gamma and its special case the χ², the uniform, and above all the normal, standardised to Z so any normal probability becomes a table lookup. It culminates in the two great limit theorems: the law of large numbers (sample means converge to the true mean) and the central limit theorem (sums and means of i.i.d. variables are approximately normal regardless of the parent), which is the reason the normal, t, χ² and F distributions dominate the inference half.
What this chapter covers
- 013.1 Density vs PMF; probability as area
- 023.2 The normalising constant and the CDF
- 033.3 Mean, median and variance of a continuous variable
- 043.4 The exponential and its memoryless property
- 053.5 The gamma, χ² and uniform families
- 063.6 The normal and standardisation to Z
- 073.7 The law of large numbers and the central limit theorem
Worked example: a piecewise density — constant, CDF and a probability
- +1(a) Normalising constant. The density must integrate to one: ∫02 cx dx = c·[x²/2]02 = 2c = 1, so c = 1/2.
- +1(b) CDF by integration. For 0 ≤ x ≤ 2, F(x) = ∫0x (1/2)t dt = x²/4; F(x) = 0 for x < 0 and 1 for x > 2.
- +1(c) A probability. P(X ≤ 1) = F(1) = 1/4.
- +1Mean. E[X] = ∫02 x·(1/2)x dx = (1/2)·[x³/3]02 = (1/2)(8/3) = 4/3.
- +1Interpret. The density rises linearly, so mass concentrates toward x = 2 and the mean 4/3 sits right of the midpoint 1 — a quick sanity check on the integral.
Key terms
- Probability density function (PDF)
- A non-negative function f(x) integrating to one, with P(a ≤ X ≤ b) = ∫abf(x)dx. Probability is area under the curve, and any single point has probability zero, so ≤ and < are interchangeable for continuous variables.
- Cumulative distribution function (CDF)
- F(x) = P(X ≤ x) = ∫−∞xf(t)dt — the running total of probability. It is non-decreasing from 0 to 1, and f = F′ recovers the density. The median solves F(m) = 0.5.
- Standardisation
- Z = (X − μ)/σ turns any normal into the standard normal N(0,1), so a probability about X becomes a lookup of Φ on the provided table. The single most-used move in the inference half.
- Central limit theorem (CLT)
- For i.i.d. Xᵢ with mean μ and finite variance σ², the standardised sample mean (X̄ − μ)/(σ/√n) converges to N(0,1) as n grows — whatever the parent distribution. It justifies normal-based inference for means and large-sample CIs and tests.
- Law of large numbers (LLN)
- The sample mean X̄ converges to the population mean μ as the sample size grows — the formal reason averaging more data sharpens an estimate. It underpins the consistency of moment-based estimators.
Continuous Random Variables and the CLT FAQ
Why is P(X = x) = 0 for a continuous variable?
Because probability is the area under the density, and a single point has zero width and therefore zero area. Only intervals carry probability. A practical consequence: for continuous variables P(X ≤ a) and P(X < a) are equal, so you never have to fuss over a strict-versus-weak inequality at an endpoint — unlike the discrete case.
When can I use the normal approximation from the CLT?
When you are working with a sum or mean of i.i.d. observations with finite variance and the sample is reasonably large. The CLT does not require the parent to be normal — that is its power — but the approximation is better for larger n and for less skewed parents. It is the licence behind large-sample confidence intervals and z-tests for means and proportions.
How do I find a median or a probability from a density?
Build the CDF first by integrating the density, then everything is a lookup: a probability P(X ≤ b) is F(b); the median solves F(m) = 0.5; a percentile pₕ solves F(x) = p. Always normalise the density (solve for the constant so it integrates to one) before computing anything — an un-normalised density gives wrong probabilities and moments.
Exam move
Drill the four-step density workflow until it is automatic: normalise (find the constant), integrate to the CDF, then read off probabilities, the median and the moments. Make standardisation a reflex — converting any normal probability to a Φ lookup is the most-used single move in the whole subject. Keep the family relationships in view (the χ² is a gamma; the exponential is a gamma with shape one) so a transformation question later resolves to a named law. Above all, internalise the CLT: state it precisely and know exactly which inference results it licenses, because the t, χ² and F machinery of the second half all trace back to it.
