MAST90105 · Methods Of Mathematical Statistics
Transformations and Sampling Distributions
Given the distribution of X, how do you find the distribution of Y = g(X)? The chapter gives three recipes and tells you when each is fastest. The CDF method works always: write FY(y) = P(g(X) ≤ y), translate the event back into X, and differentiate to get the density. The Jacobian (change-of-variables) method is the shortcut for a monotone g, multiplying fX by |dx/dy|. The MGF method identifies the result by name — compute the MGF of Y and match it to a known family, which is unbeatable for sums of independents. The payoff is the manufacture of the sampling distributions that drive all of inference: for a normal sample, the sample mean X̄ is normal and (n−1)S²/σ² is χ², independent of X̄; their ratios produce the t and F distributions. The whole Z → t → χ² → F family that you keep meeting in confidence intervals and tests is built right here.
What this chapter covers
- 015.1 The CDF method — always works
- 025.2 The Jacobian method for monotone transformations
- 035.3 The MGF method — naming the result
- 045.4 The sampling distribution of X̄ for normal data
- 055.5 (n−1)S²/σ² is χ², independent of X̄
- 065.6 Building t and F; the Z → t → χ² → F family
Worked example: the CDF method on Y = X²
- +1Name the method. g(x) = x² is monotone on [0,1], so either the CDF or the Jacobian method works; we use the CDF method as the all-purpose recipe.
- +1Support. As x ranges over [0,1], y = x² ranges over [0,1], so fY is supported on 0 ≤ y ≤ 1.
- +1CDF of Y. FY(y) = P(X² ≤ y) = P(X ≤ √y) = FX(√y) = √y for 0 ≤ y ≤ 1 (since FX(x) = x on [0,1]).
- +1Differentiate. fY(y) = d/dy(√y) = 1/(2√y) for 0 < y ≤ 1.
- +1Check. ∫01 1/(2√y) dy = [√y]01 = 1, so fY is a valid density. It blows up near 0, reflecting how squaring crowds small values toward zero.
- +0Jacobian cross-check. x = √y, |dx/dy| = 1/(2√y), and fY = fX(√y)|dx/dy| = 1·1/(2√y) — same answer.
Key terms
- CDF method
- Find the law of Y = g(X) by writing FY(y) = P(g(X) ≤ y), rewriting the event in terms of X, evaluating through FX, and differentiating to get fY. It works for any g, monotone or not — the all-purpose recipe.
- Jacobian method
- For a one-to-one monotone g, fY(y) = fX(x)·|dx/dy| evaluated at x = g⁻¹(y). It is the fastest route when g is invertible, but needs care with the support and with non-monotone maps.
- MGF method
- Compute MY(t) and match it to a known family’s MGF to name the distribution of Y. Combined with the product rule for sums of independents, it identifies the law of a sum without any integration.
- Chi-square / sample variance
- For an i.i.d. normal sample, (n−1)S²/σ² ~ χ²n−1 and is independent of the sample mean X̄. This is the building block for inference about a variance and for the t-statistic.
- t and F distributions
- A standard normal over the square root of an independent χ²/df gives a t; a ratio of two independent χ²/df gives an F. These constructions are why t appears for an unknown-σ mean and F for a ratio of variances.
Transformations and Sampling Distributions FAQ
Which transformation method should I use?
Use the CDF method as your default — it works for any g, including non-monotone maps, where you split the event into pieces. Use the Jacobian method when g is one-to-one and monotone, for speed. Use the MGF method when you need to name the result, especially for a sum of independent variables, since matching MGFs identifies the family with no integration. Knowing which to reach for is itself examined.
Where do the t, chi-square and F distributions actually come from?
They are manufactured from a normal sample. The sample variance gives (n−1)S²/σ² ~ χ², independent of the mean. A standard normal divided by the square root of an independent χ²/df is a t — which is why the t-statistic appears when σ is unknown. A ratio of two independent χ²/df is an F, used for comparing variances. The whole family descends from Z by these transformations.
Why does the density of Y = X² blow up near zero?
Because squaring compresses small values: a wide band of small x maps into a narrow band of small y, so probability piles up there, and the density 1/(2√y) is large near zero. It is still a valid density — it integrates to one — because the singularity is integrable. Always check that your transformed density integrates to one and respects the transformed support.
Exam move
Make the CDF method your reliable default and practise it on both monotone and non-monotone g until splitting the event is routine; then learn the Jacobian shortcut for monotone maps and the MGF method for naming sums. The highest-value target is the construction of the sampling distributions: be able to state that X̄ is normal and (n−1)S²/σ² is χ² and independent of X̄, and to assemble t and F from a Z and independent χ²s. Carry the Z → t → χ² → F map on your A4 sheet — it explains every test statistic in the inference half.
