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Question
I have a joint probability mass function: p(x,y)=2xy/108 (for x=1,3,5; y = 1,2,3)

How do I determine the marginal distributions for X and Y (table or equation)?

How do I determine E(X) and V(X)?

What is the conditional probability, P(X = 5 given Y = 3)?

Are X and Y therefore independent?

Answer
Ricardo,

Another HW problem (?!). The marginal distribution for x is obtained by summing p(x,y) over y for each value of x. In the table below, I used Excel to do the summing. The rightmost column is the marginal distribution for x. The marginal distribution for y is done in the same way, i.e., by summing over x; see bottom row.

probability calcs      p=2xy/108      
  y         
x   1          2          3          sum=x marg
1   0.018518519   0.037037037   0.055555556   0.111111111
3   0.055555556   0.111111111   0.166666667   0.333333333
5   0.092592593   0.185185185   0.277777778   0.555555556

sum=y marg   0.166666667   0.333333333          0.5          1

        1   

You should realize that summing over both x and y (all entries) should give a value of 1, which is the condition used in all probability analysis that the total probability is 1. You should try this. I also summed over the 2 marginal probabilities separately which should also equal 1.

To determine E(X), just multiply each of the (3) marginal distribution entries for x by the corresponding value of x and sum up (this is the definition of an expected value, which you should have in your textbook).

The conditional probability for P(5,3) is just the value of the entry at x=5 and y=3, namely 0.278.

Since the probability for a given x depends on the value of y, i.e., P(5,3) ≠ P(5,1) ≠ P(5,2), and vice versa, I'd say they are not independent.

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