14.3. Marginal Distributions

Try these first

1. 1.

   What is the formula for the marginal PMF of X in a discrete joint distribution?

   Hint

   Think of summing over the probabilities in the joint PMF.

2. 2.

   In continuous distributions, what do we do to find the marginal PDF of X?

   Hint

   Integration will help you collapse the other variable.

3. 3.

   What does a marginal distribution refer to in probability theory?

   • The overall distribution of multiple variables
   • The distribution of a single variable
   • The joint distribution of two variables

   Hint

   Focus on the term 'marginal' which indicates focusing on one aspect.

4. 4.

   True or False: Marginal distributions can only be derived from discrete random variables.

   • True
   • False

   Hint

   Consider the definitions of marginals for different types of random variables.

5. 5.

   In a joint distribution of two dice rolls, what is the marginal distribution of one die? Provide a detailed explanation of your process.

   Hint

   Start with a joint probability table, then aggregate relevant entries.

6. 6.

   Given a joint PDF of f(x,y) = 2x + 3y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1, derive the marginal PDFs and verify if they support independence.

   Hint

   Evaluate the integrals correctly and then set up their products.