15.3. Discrete Case

Try these first

1. 1.

   What is the marginal pmf of X if the joint pmf p(x,y) = 0.2 for (x=0,y=1) and p(0,2) = 0.3?

   Hint

   Remember to sum probabilities across possible values of Y.

2. 2.

   Define marginal distribution in your own words.

   Hint

   Think about focusing on one variable only.

3. 3.

   What is the formula for the marginal pmf of X?

   • p(x) = ∑ p(x,y)
   • p(x) = ∫ p(x,y) dy
   • p(x) = p(x,y) - p(y)

   Hint

   Focus on how we isolate one variable.

4. 4.

   True or False: Marginal distributions can always be used to reconstruct the joint distribution.

   • True
   • False

   Hint

   Consider if independence is necessary for reconstruction.

5. 5.

   Consider the joint pmf for two discrete random variables: p(0,0)=0.3, p(0,1)=0.1, p(1,0)=0.4, and p(1,1)=0.2. Calculate the marginal distribution for both variables.

   Hint

   Sum probabilities for each variable correctly.

6. 6.

   If the two variables X and Y are independent, deduce the relationship of their joint pmf to their marginal pmfs mathematically.

   Hint

   Reflect on definitions of independence.