Discussion
Let X a \(p\)-component random vector is partitioned into \(X^{(1)}\) and \(X^{(2)}\) where \(X^{(1)}\) has \(q\)-components and \(X^{(2)}\) has \((p - q)\) components and \(X \sim N_{p}(\mu, \Sigma)\). If \(\mu\) and \(\Sigma\) are also partitioned as of X, then the variance covariance matrix of the conditional distribution of \(X^{(1)}\) given \(X^{(2)}\) is :
Answer: B
The conditional covariance matrix is \(\Sigma_{11} - \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}\).
For a partitioned multivariate normal, the conditional distribution of \(X^{(1)}\) given \(X^{(2)} = x^{(2)}\) is normal with mean \(\mu^{(1)} + \Sigma_{12}\Sigma_{22}^{-1}(x^{(2)} - \mu^{(2)})\) and covariance \(\Sigma_{11} - \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}\), the Schur complement of \(\Sigma_{22}\).
Conditioning can only reduce variability, so the correction is subtracted, ruling out the two options with a plus sign. The options built from \(\Sigma_{22}\) give the conditional covariance of \(X^{(2)}\) given \(X^{(1)}\), the other way round.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development