Discussion
Let \((X, Y) \sim N(\underline{0}, \Sigma)\) where \(\Sigma = \begin{bmatrix} 1 & \rho \\ \rho & 1 \end{bmatrix}\), \(\rho \neq 0\) and
\[ W = \frac{X - Y}{X + Y}\sqrt{\frac{1 + \rho}{1 - \rho}}. \]
Then which of the following statements is not correct ?
Answer: D
The statement that is not correct is \(W \sim \chi_{1}^{2}\).
\(X - Y \sim N(0, 2(1 - \rho))\) and \(X + Y \sim N(0, 2(1 + \rho))\), and \(\mathrm{Cov}(X - Y, X + Y) = \mathrm{Var}X - \mathrm{Var}Y = 0\), so being jointly normal they are independent (C). Standardising,
\[ \begin{aligned} W &= \frac{(X - Y)/\sqrt{2(1 - \rho)}}{(X + Y)/\sqrt{2(1 + \rho)}} \\ &= \frac{Z_{1}}{Z_{2}} \end{aligned} \]
a ratio of independent standard normals, so W is standard Cauchy. Then \(W^{2} = Z_{1}^{2}/Z_{2}^{2} \sim F_{1,1}\) (A), and so is \(1/W^{2}\) (B). W takes negative values, so it cannot be chi-square.
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