I understand the algebraic proof of the linearity of expectation, but I cannot grasp the intuition, especially when the random variables are not independent. It is even counterintuitive as you cannot simply add the probabilities of dependent random variables, but it does not affect on the expected value.

Can anyone explain the intuition behind the algebraic formula? How to think about it?

  • $\begingroup$ such argument gives an algebraic proof, but it doesn't explain the reason behind the property $\endgroup$ – guser Dec 4 '17 at 20:30
  • $\begingroup$ This provides an intuitive explanation. $\endgroup$ – Remy Dec 4 '17 at 20:37

This site https://brilliant.org/wiki/linearity-of-expectation/ was using the idea of total rainfall. How much rain do we expect to get over the next 3 days? It should be the amount we expect to get over the first day, plus the second, and then third day's expected values. Even though it's not independent (for instance a big storm can take more than 1 day to blow through), our system must be able to add up the individual expected values to get the larger expected value or the system isn't working right.

  • $\begingroup$ so what you mean is that even in counting the expected amount of rain for one day we already take into account the dependence on the amount of rain on the other day? $\endgroup$ – guser Dec 4 '17 at 20:52
  • $\begingroup$ The added variables can be dependent or independent and should still hold true. If you did have an accurate way to get expected amounts of rain for each day, then that method would 'know about' the dependence between days. $\endgroup$ – user78090 Dec 4 '17 at 21:16

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