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Part CCSIR NET June 2024a-singular-covariance-means-one-component

A singular covariance means one component

Let X=(X1,X2)TX = (X_{1}, X_{2})^{T} be a bivariate random vector with covariance matrix ∑=[[1,2],[2,2]]\sum = [[1, \sqrt{2}], [\sqrt{2}, 2]]. Which of the following statements are true?

  1. A.The first principal component based on ∑\sum explains exactly 90% of the total variability
  2. B.The second principal component based on ∑\sum explains exactly 10% of the total variability
  3. C.sup{aT∑a:a∈R2a^{T}\sum{}a : a \in \mathbb{R}^{2} and aᵀa = 1} = 3✓ Correct
  4. D.The first principal component based on ∑\sum is (1/3)(X1+2X2)(1/\sqrt{3})(X_{1} + \sqrt{2X_{2}})✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests invariants don't determine.

See pricing

50 are analysed free — try those first.

The trap it tests

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

Drill statements like this

Related counterexample: If every marginal is normal then the vector is multivariate normal

More on this topic

The chapter behind this: Multivariate normal and Wishart — free to read

From Linear Models and Multivariate › Multivariate normal distribution

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