17.5. Examples

Try these first

1. 1.

   What does it mean for random variables to be independent?

   Hint

   Think of the dice examples we discussed.

2. 2.

   How is the joint PMF represented for discrete random variables?

   Hint

   Refer to the table format presented in our example.

3. 3.

   What indicates that two random variables are independent?

   • P(X,Y) = P(X) * P(Y)
   • P(X,Y) < P(X) * P(Y)
   • P(X,Y) > P(X) * P(Y)

   Hint

   Anchor this to the conditional probabilities we discussed.

4. 4.

   In discrete random variables, which function describes the joint probabilities?

   • Marginal Probability Function
   • Joint Probability Mass Function
   • Mutual Information

   Hint

   Check the terminology we used before.

5. 5.

   Given X and Y with the joint PDF f(x,y) = e^(-x) * e^(-y), derive whether X and Y are independent.

   Hint

   Calculate the marginal PDFs first.

6. 6.

   Create a joint PMF for X and Y that showcases independence and calculate all relevant probabilities.

   Hint

   Make sure distributions sum to 1 and check independence condition afterward.