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

The Correct Option is B, C, D

This question examines the properties of independent normal random variables and their linear combinations.

Understanding the Distributions: Let \(X\) and \(Y\) be independent random variables. \(X\) follows a normal distribution \(N(0, \sigma^2_1)\) and \(Y\) follows \(N(0, \sigma^2_2)\).

Independence implies:

• The covariance \(cov(X, Y) = 0\).

Evaluating Independence of Linear Combinations: For two linear combinations \(a_1X + b_1Y\) and \(a_2X + b_2Y\) to be independent, their covariance should be zero.

The covariance is calculated as:

\(cov(a_1X + b_1Y, a_2X + b_2Y) = a_1a_2var(X) + b_1b_2var(Y) + (a_1b_2 + a_2b_1)cov(X, Y)\)

Given that \(cov(X, Y) = 0\), the formula simplifies to:

\(a_1a_2\sigma^2_1 + b_1b_2\sigma^2_2 = 0\)

Analyzing Each Option:

Option 1: \(X + Y\) and \(X - Y\)

Calculate: \(cov(X + Y, X - Y) = var(X) - var(Y) = \sigma^2_1 - \sigma^2_2 \neq 0\). They are not independent.

Option 2: \(2X + Y\) and \(X - Y\) when \(2\sigma^2_1 = \sigma^2_2\)

Calculate: \(cov(2X + Y, X - Y) = 2\sigma^2_1 - \sigma^2_2 = 0\) (since given \(2\sigma^2_1 = \sigma^2_2\))

Thus, they are independent under the condition.

Option 3: \(X + Y\) and \(X - Y\) are identically distributed.

Since they both have the same variance: \((var(X+Y) = \sigma^2_1 + \sigma^2_2 = var(X-Y))\), they are identically distributed.

Option 4: \(X + Y\) and \(2X - Y\) when \(2\sigma^2_1 = \sigma^2_2\)

Calculate: \(cov(X + Y, 2X - Y) = 2\sigma^2_1 - \sigma^2_2 = 0\) (since given \(2\sigma^2_1 = \sigma^2_2\))

Hence, they are independent under the condition.

Conclusion: The true statements are:

• \(2X + Y\) and \(X - Y\) are independent if \(2\sigma^2_1 = \sigma^2_2\)
• \(X + Y\) and \(X - Y\) are identically distributed
• \(X + Y\) and \(2X - Y\) are independent if \(2\sigma^2_1 = \sigma^2_2\)