Teaching CSIR NET Mock Test Series Mathematical Science Statistics & Exploratory Data Analysis Analysis of Variance and Covariance
Let X = \(\rm \begin{pmatrix}X_1\\\ X_2\end{pmatrix}\) be a bivariate random vector with covariance matrix \(\rm \Sigma=\rm \begin{pmatrix}1&\sqrt2\\\ \sqrt2&2\end{pmatrix}\)
Which of the following statements are true?
1
The first principal component based on Σ explains exactly 90% of the total variability
2
The second principal component based on Σ explains exactly 10% of the total variability
3
\(\sup \left\{\underline{a}^T\Sigma \underline{a}:\underline{a}\in \mathbb{R}^2\text{ and }\underline{a}^T\underline{a}=1\right\}=3\)
4
The first principal component based on Σ is \(\rm \frac{1}{\sqrt3}(X_1+\sqrt2X_2)\)