Given the joint probability distribution table of $X$ and $Y$, write $XY$

Question

I have the two random dependent variables, $X$ and $Y$, whose distribution is given by a joint probability distribution table:

$$\begin{array}{c|cccc} X,Y&1&3&6&9\\ \hline 2&0.11&0.05&0.20&0.08\\ 3&0.20&0.02&0.00&0.10\\ 7&0.00&0.05&0.10&0.09\\ \end{array}$$ I need to write $Z=XY$ and find the probability distribution of $Z$. Sorry if this is an obvious question, I'm just really struggling to understand what I need to do.

Answer 1

Compute $P(Z=z)$ where $z$ is some number as follows:

1. Find all the possible ways that $X$ and $Y$ can multiply to get $z$ from the table.
2. Get their probabilities from the table.
3. Add them up.

For instance if we want $P(Z=9)$

1. The only way we can get $XY=9$ is when $X=3$ and $Y=3/$
2. The probability $X=Y=3$ is $0.02.$
3. There is only one thing to add up, so we have $P(Z=9)=0.02.$

If we want $P(Z=5)$ all we need do is observe from the table that there is no way to multiply $X$ and $Y$ to get $5,$ so $P(Z=5)=0.$

The answer probably expects that you figure out all $z$ that can be formed and write down the probability of each in the table. So I've done one row $(9,0.02)$ for you.