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

A singular covariance means one component

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

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

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.

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A singular covariance means one component

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