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Part BCSIR NET June 2025the-marginal-correlation-is-one-half-and-all-of-it-is-the-shared-x

The marginal correlation is one half and all of it is the shared x

Suppose (X1,X2,X3)T(X_{1}, X_{2}, X_{3})^{T} ~ N3(0,I3)N_{3}(0, I_{3}), where 0 is the zero mean vector and I3I_{3} is the 3 × 3 identity matrix, and (X,Y,Z)T=A(X1,X2,X3)T(X, Y, Z)^{T} = A(X_{1}, X_{2}, X_{3})^{T} where A has rows (3, 0, 0), (2, 2, 0) and (4, 0, 4). Then the partial correlation coefficient ρYZ.X\rho_{YZ.X} is

  1. A.1/2
  2. B.2/3
  3. C.3/4
  4. D.0✓ 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 dependence misread.

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50 are analysed free — try those first.

The trap it tests

Dependence misread

Independence, pairwise vs mutual, or correlation vs dependence.

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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