17.7. Tests and Theorems Related to Independence

Try these first

1. 1.

   What does it mean for two random variables to be independent?

   Hint

   Look for definitions related to random variables.

2. 2.

   If Cov(X,Y) = 0, what can we infer about the relationship between X and Y?

   Hint

   Think about the implications of zero covariance.

3. 3.

   True or False: Independence of random variables means they have covariance of zero.

   • True
   • False

   Hint

   Remember the distinction between independence and correlation.

4. 4.

   What indicates that two variables are independent?

   • Zero covariance
   • Non-zero covariance
   • Zero mutual information
   • Same variance

   Hint

   Link mutual information to independence.

5. 5.

   Two random variables X and Y are defined with specific joint distributions. Calculate Cov(X,Y) and argue whether they might be independent.

   Hint

   Use the definitions and calculate accordingly.

6. 6.

   Find a real-world scenario in communication systems where mutual information would be integral. Discuss its implications on the system design.

   Hint

   Consider how signals and noise interactions might offer insights.