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Part BCSIR NET December 2025correlation-is-scale-invariant-work-directly-with-the-given-covariance-matrix

Correlation is scale invariant work directly with the given covariance matrix

Let X=(X1,X2,X3)TX = (X_{1},X_{2},X_{3})^{T} be a 3×1 random vector with E(X) = (3,2,1)ᵀ and Cov(X)=∑Cov(X) = \sum, with rows (3,−2,0), (−2,3,−2), (0,−2,3). Suppose that Y=(Y1,Y2,Y3)T=(1/3)XY = (Y_{1},Y_{2},Y_{3})^{T} = (1/\sqrt{3})X. Then the value of the multiple correlation coefficient between Y1Y_{1} and (Y2,Y3)(Y_{2},Y_{3}) equals

  1. A.2/52/\sqrt{5}✓
  2. B.2/5\sqrt{2/5}
  3. C.2/32/\sqrt{3}
  4. D.1/31/\sqrt{3}

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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50 are analysed free — try those first.

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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