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

Principal component loadings are unit norm

Let $X_{0}, X_{1}, \dots, X_{p} (p \geq 2)$ be independent and identically distributed random variables with mean 0 and variance 1. Suppose $Y_{i} = X_{0} + X_{i}, i = 1, \dots, p$ . The first principal component based on the covariance matrix of $Y = (Y_{1}, \dots, Y_{p})^{T}$ is

A. $(1/ p) \sum_{i}$ ₌ $_{1}$ ᵖ $Y_{i}$ ✓
B. $(1/ p) \sum_{i}$ ₌ $_{1}$ ᵖ $Y_{i}$
C. $p \sum_{i}$ ₌ $_{1}$ ᵖ $Y_{i}$
D. $\sum_{i}$ ₌ $_{1}$ ᵖ $Y_{i}$

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.

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Principal component loadings are unit norm

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