Let \(X_0, X_1, \ldots, X_p ( p \geq 2 )\) be independent and identically distributed random variables with mean 0 and variance 1. Define \(Y_i = X_0 + X_i \) for i = 1, ...., p. The covariance matrix of \(Y = (Y_1, \ldots, Y_p)^T\) is denoted by Σ. What is the first principal component based on Σ?
1
\( \frac{1}{p} \sum_{i=1}^p Y_i \)
2
\( \sqrt{p} \sum_{i=1}^p Y_i \)
3
\(\frac{1}{\sqrt{p}} \sum_{i=1}^p Y_i \)
4
\( \sum_{i=1}^p Y_i \)