[Math] Probability distribution of the product of two dependent variables

probabilityrandom variables

The following joint probability distribution exists for two dependent random variables

               Y
           1    3    6   9
       2 0.11 0.05 0.20 0.08
    X  3 0.20 0.02 0.00 0.10
       7 0.00 0.05 0.10 0.09

How do I find $\mathbf{P}(Z = XY)$?

I'm not sure if I'm supposed to sum across all rows/columns, then find the products, or just take them as it is.
For example:

Is $\mathbf{P}(Z = 63) = 0.09$, or $\mathbf{P}(X=7) \cdot \mathbf{P}(Y=9)$, that is $0.24 \cdot 0.27 = 0.065$?

Best Answer

$P(Z=XY)$ is just a look-up. No multiplication involved. You will have to deal with multiple $X,Y$ pairs which give the same $Z$ in which case, the probabilities are additive.

$$P(Z=63)=0.09$$

A good way of testing whether ANYTHING is a valid PDF is to see if it integrates/sums up to 1.

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