Discussion
Suppose \(X_{1}, X_{2}, X_{3}\) and \(X_{4}\) are independent and identically distributed tri-variate random vectors having normal distribution. Then the distribution of \(a_{1}X_{1} + a_{2}X_{2} + a_{3}X_{3} + a_{4}X_{4}\), where \(a_{1}, a_{2}, a_{3}, a_{4}\) are real numbers, is :
Answer: C
The distribution is trivariate normal.
Each \(X_{i}\) is a 3-dimensional normal vector, and they are independent, so the stacked 12-dimensional vector is multivariate normal. The combination \(\sum a_{i}X_{i}\) is a linear transformation of it into \(\mathbf{R}^{3}\), and linear transformations of multivariate normal vectors are normal. Its mean is \(\sum a_{i}\mu\) and its covariance \((\sum a_{i}^{2})\Sigma\).
The result is still a 3-component vector, so it is neither univariate nor four-variate; the number 4 counts the vectors summed, not the dimension. And it is normal, not 'not normal'.
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