Question

Let X and Y be two independent random variables having N(0, $σ^2_1$) and N(0, $σ^2_2$) distributions, respectively, where 0 < σ1 < σ2. Then which of the following statements is/are true ?

• A. X + Y and X - Y are independent
• B. 2X + Y and X - Y are independent if $2σ^2_1=σ^2_2$
• C. X + Y and X - Y are identically distributed
• D. X + Y and 2X - Y are independent if $2σ^2_1=σ^2_2$

Answer 1

Answer: (B), (C), (D)

2X + Y and X - Y are independent if $2σ^2_1=σ^2_2$

Answer 2

The correct option is (B) : 2X + Y and X - Y are independent if $2σ^2_1=σ^2_2$, (C) : X + Y and X - Y are identically distributed and (D) : X + Y and 2X - Y are independent if $2σ^2_1=σ^2_2$.