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If I have two independent random variables, how should I calculate the product of them?

For example: enter image description here

How should I calculate XY?

Thank you.

1 Answers 1

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You can just make a table

$$\begin{bmatrix} X \text{\\} Y && 1 && 4 && 6 \\ 2 && .12 && .04 && .04 \\ 3 && .18 && .06 && .06 \\ 5 && .30 && .10 && .10 \end{bmatrix}$$

So $P(XY= x\cdot y)=P(X=x)P(Y=y)$ and you get the table from there. In general if you have any overlap of products, you would need to add them together, in this case there are no duplicates, so you just have that table, but if eg $X$ had a $4$ value and $Y$ had a $2$ value, then $X=4, Y=1$ and $X=Y=2$ would both contribute to $P(XY=4)$.

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    Thanks for the answer. I would like to know if there is any way to transform the table or directly calculate the product as a random variable (a matrix with two lines just like those in the original post)? – 2017-02-03
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    @TomaRadu-Petrescu not in general. You **can** let $A_X, A_Y$ be the vectors of values for $X,Y$ and consider the matrix $A_X\otimes A_Y^T$, but even then if you have repeats like the caveat I gave in my post you would have to manually reduce the matrix. – 2017-02-03