Multiply mean values

Question

I was wondering if its allowed to multiply two mean values. I thought of the following: $$y = x_1 * x_2$$ They values $x_1$ and $x_2$ are from different sample sets and are not related. They only have to same number of samples in them.
I think its wrong to calculate $\bar{y}$ like this: $$\bar{y}=\bar{x_1}*\bar{x_2} $$ Because: $$ \bar{y} = \frac{1}{n}\sum x_{1i} * \frac{1}{n}\sum x_{2i} = \frac{1}{n^2}\sum x_{1i}x_{2i} $$ Am I right ?

or does it resolve to this and it is allowed? $$ \bar{y} = \frac{1}{n}\sum x_{1i} * \frac{1}{n}\sum x_{2i} = \frac{1}{n}\sum x_{1i}x_{2i} $$

Answer 1

$$\sum x_{1i} \times \sum x_{2i} \neq \sum x_{1i}x_{2i}$$

As a simple example, consider

$$(1 + 2 + 3) \times (4 + 5 + 6) \neq 1\cdot4 + 2\cdot 5 + 3 \cdot 6$$

Answer 2

two sums do not multiply like you did it. Look up Cauchy product, $(x_1+x_2)*(y_1+y_2)\ne x_1y_1+x_2y_2$