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Part BCSIR NET June 2024principal-component-loadings-are-unit-norm

Principal component loadings are unit norm

Let X0,X1,…,Xp(p≥2)X_{0}, X_{1}, \dots, X_p (p \ge 2) be independent and identically distributed random variables with mean 0 and variance 1. Suppose Yi=X0+Xi,i=1,…,pY_{i} = X_{0} + X_{i}, i = 1, \dots, p. The first principal component based on the covariance matrix of Y=(Y1,…,Yp)TY = (Y_{1}, \dots, Y_p)^{T} is

  1. A.(1/p)∑i(1/\sqrt{p}) \sum_{i}₌1_{1}ᵖ YiY_{i}✓ Correct
  2. B.(1/p)∑i(1/p) \sum_{i}₌1_{1}ᵖ YiY_{i}
  3. C.p∑i\sqrt{p} \sum_{i}₌1_{1}ᵖ YiY_{i}
  4. D.∑i\sum_{i}₌1_{1}ᵖ YiY_{i}

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 execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

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

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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