Question

Let the discrete random variables \( X \) and \( Y \) have the joint probability mass function 

Which of the following statements is (are) TRUE? 
 

• A. The marginal distribution of \( X \) is Poisson with parameter 2
• B. The marginal distribution of \( X \) and \( Y \) are independent
• C. The joint distribution of \( X \) and \( X + \sqrt{Y} \) is independent
• D. The random variables \( X \) and \( Y \) are independent

Hint:

When the joint distribution factorizes as the product of marginal distributions, the random variables are independent.

Answer 1

The Correct Option is B, C, D

Step 1: Use the joint probability mass function. 
The given joint PMF suggests that \( X \) and \( Y \) are independent, since the joint distribution factorizes as a product of the marginal distributions. The marginal distributions of \( X \) and \( Y \) are Poisson with parameter 2. Thus, \( X \) and \( Y \) are independent. 
Step 2: Analyzing the options. 
(A) The marginal distribution of \( X \) is Poisson with parameter 2: This is true. The marginal distribution of \( X \) is Poisson with parameter 2, as shown by the given joint PMF. 
(B) The marginal distribution of \( X \) and \( Y \) are independent: This is true. The joint distribution factors as the product of the marginal distributions, implying that \( X \) and \( Y \) are independent. 
(C) The joint distribution of \( X \) and \( X + \sqrt{Y} \) is independent: This is false. The variables \( X \) and \( X + \sqrt{Y} \) are not independent because \( X \) influences \( X + \sqrt{Y} \). 
(D) The random variables \( X \) and \( Y \) are independent: This is true, as shown in Step 1. 
Step 3: Conclusion. 
The correct answer is D, as \( X \) and \( Y \) are independent random variables.