So I know the linearity of expectation allow us to adds / subtracts Expectation (i.e. sums of expectation equals to the expectation of the sums)
However, I am thinking about the following:
Let $X$ be the number that appears on a d6 die and $Y$ be the number of a d8 die.
Then the expected sum of the two die will be $E[X+Y] = E[X] + E[Y] = 8$
But as far as I know, multiplication is not linear, therefore if we are asking the expected product of the two die $E[XY]$, it shouldn't be $E[X]E[Y]$, though it seems to be yielding the same result.
So my question here will be, when does linearity of expectation works outside of the scope of addition and subtraction? Or how do we know if that will work or not?
If $X,Y$ are independent i.e. $P(X=x\text{ and }Y=y)=P(X=x)\times P(Y=y)$, then the expectation of the product $XY$ is the product of their expectations.
This is because$$E[XY]=\sum_{x\in D_x}\sum_{y\in D_y} xyP(X=x\text{ and }Y=y)$$where $D_x$ denotes the sample space (domain) of $X$ and similarly for $D_y$. For independent $X,Y$, this becomes$$\sum_{x\in D_x}\sum_{y\in D_y}xyP(X=x)P(Y=y)=\sum_{x\in D_x}xP(X=x)\times\sum_{y\in D_y}yP(Y=y)=E[X]E[Y]$$