15.6. Worked Example (Continuous Case)

Try these first

1. 1.

   What does the joint pdf represent in probability theory?

   Hint

   Think about how two random variables can interact.

2. 2.

   What is the integral used for when finding marginal distributions?

   Hint

   Recall the definition of marginalization.

3. 3.

   What does the marginal distribution of a variable express?

   • The distribution of the entire population
   • The distribution of a single variable
   • ignoring others
   • Only the joint probability
   • The probability distribution over a range of data

   Hint

   Think of how we analyze one aspect at a time.

4. 4.

   True or False: You can always reconstruct joint pdfs from marginal distributions if the variables are dependent.

   • True
   • False

   Hint

   Consider the definition of dependency.

5. 5.

   Consider the joint pdf f(x,y) = 4x^2y^2 for 0 < x < 1, 0 < y < 1. Derive both marginal distributions.

   Hint

   Remember to integrate over the other variable to calculate each marginal.

6. 6.

   Given a joint pdf f(x,y) = xy for 0 < x < 2 and 0 < y < 2, analyze its marginal distributions and check if the variables are independent.

   Hint

   Revisit the definition of independence and how it applies to our findings.