Question

Let (X, Y) be a discrete random vector. Then which of the following statements is/are true ?

• A. If X and Y are independent, then X² and |Y| are also independent.
• B. If the correlation coefficient between X and Y is 1, then P(Y = aX + b) = 1 for some a, b ∈ \(\R\)
• C. If X and Y are independent and E[(XY)²] = 0, then P(X = 0) = 1 or P(Y = 0) = 1
• D. If Var(X) = 0, then X and Y are independent

Answer 1

The Correct Option is A, B, C, D

To determine which statements are true, let's analyze each option carefully within the context of probability and statistics.

1. Statement: If \( X \) and \( Y \) are independent, then \( X^2 \) and \(|Y|\) are also independent.
   Explanation: Independence of random variables \( X \) and \( Y \) means that for any functions of these variables, say \( f(X) \) and \( g(Y) \), are also independent. Therefore, \( X^2 \) is a function of \( X \) and \(|Y|\) is a function of \( Y \). Hence, they remain independent.
   Conclusion: This statement is true.
2. Statement: If the correlation coefficient between \( X \) and \( Y \) is 1, then \( P(Y = aX + b) = 1 \) for some \( a, b \in \mathbb{R} \).
   Explanation: A correlation coefficient of 1 implies perfect linear relationship between \( X \) and \( Y \). Thus, \( Y = aX + b \) surely holds for some constants \( a \) and \( b \).
   Conclusion: This statement is true.
3. Statement: If \( X \) and \( Y \) are independent and \( E[(XY)^2] = 0 \), then \( P(X = 0) = 1 \) or \( P(Y = 0) = 1 \).
   Explanation: The expectation \( E[(XY)^2] = 0 \) indicates that the squared product of \( X \) and \( Y \) is zero almost surely. This can only occur when either \( X = 0 \) or \( Y = 0 \) with probability 1.
   Conclusion: This statement is true.
4. Statement: If \( \text{Var}(X) = 0 \), then \( X \) and \( Y \) are independent.
   Explanation: Variance being zero means \( X \) is a constant variable, say \( X = c \). A constant variable is independent of any other variable, as there is no randomness or variable nature associated with it.
   Conclusion: This statement is true.

Final Answer: All statements are correct based on the statistical properties and definitions stated above.