14.8. Example Problems

Try these first

1. 1.

   What is the marginal PMF of X for the discrete distribution given?

   Hint

   Sum the probabilities where X equals 0.

2. 2.

   How can you check if two random variables X and Y are independent?

   Hint

   Use the joint PMF for verification.

3. 3.

   What does the joint probability mass function define?

   • Probability of single outcomes
   • Probability for pairs of random variables
   • Probability for mixtures

   Hint

   Think about how they relate to each other.

4. 4.

   True or False: Marginal distributions can always be derived from joint distributions.

   • True
   • False

   Hint

   Recall the method of finding marginals.

5. 5.

   Consider a random variable Z defined with a joint PMF across three dimensions. Determine the marginal PMF for Z and check if it influences the independence of the two other dimensions.

   Hint

   Break the problem down dimension by dimension!

6. 6.

   Given a continuous distribution defined in a non-standard region, derive its joint PDF and confirm its validity before calculating marginal PDFs.

   Hint

   Focus on region boundaries for proper limits!