17.4. Mathematical Conditions for Independence

Try these first

1. 1.

   What is the mathematical condition for independence between two discrete random variables?

   Hint

   Look at the relationship between joint and marginal probabilities.

2. 2.

   How do we express independence in continuous random variables?

   Hint

   Focus on how we relate joint PDF to marginal PDFs.

3. 3.

   Which of the following is TRUE for independent random variables?

   • P(X
   • Y) = P(X) + P(Y)
   • P(X
   • Y) = P(X) * P(Y)
   • f(x
   • y) = f(x) + f(y)

   Hint

   Think about what independence mathematically implies.

4. 4.

   True or False: If two random variables are independent, their covariance is always zero.

   • True
   • False

   Hint

   Consider how correlation and independence relate.

5. 5.

   A joint PMF of two discrete variables X and Y is provided. Calculate the marginal distributions and test if X and Y are independent. If X has values 1, 2 with corresponding probabilities, and Y has values 3, 4, write out the joint PMF table.

   Hint

   Don't forget to sum across both variables carefully.

6. 6.

   Given a continuous joint PDF f(x, y) = xy for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1, derive the individual PDFs and check independence.

   Hint

   Set up correct limits for integration and pay attention to normalization.