Based on the following article: https://brilliant.org/wiki/linearity-of-expectation/
We know that:
Linearity of expectation can be applied to dependent random variables.
"The digits 1,2,3 and 4 are randomly arranged to form two two-digit numbers,
AB and CD. For example, we could have AB = 42 and CD = 13. What is the expected value of AB*CD?"
Why can't we just multiply E[AB]*E[CD] if it's ok to apply Linearity of expectation on dependant variables?
Thanks
Best Answer
As mentioned by @lulu and in the article, linearity of expectation is an additive property, not a multiplicative one.
By the way, if you were wondering how to solve the question you posed, note that there are $12$ different choices of $AB$ and $2$ choices of $CD$ for each choice of $AB.$ However, because of the commutative property of multiplication we only need to consider $12$ choices. These choices correspond to $(AB,CD)=(12,34),(12,43),(21,34),(21,43),(13,24),(13,42),(31,24),(31,42),(14,23),(14,32),(41,23),(41,32).$
Thus, the expected value of $AB\cdot CD$ is $705 \dfrac{5}{6}.$ Though @CalvinLin's solution is more efficient because it exploits the symmetry in the problem.
However your method would give $756.25.$