For linearity of expectation to work, do the random variables have to be from the same experiment?
S there are two random variables X and Y on the same sample space, which assigns real values $X(s)$ and $Y(s)$ to to very outcome $s$ of an experiment, does that mean that the $P(s)$ has to be the same?
Linearity of expectation presupposes that you can form linear combinations of the random variables. So there has to be some probability space on which all the relevant random variables are defined. The variables could, however, come from entirely separate experiments (e.g., you roll a die and I flip a coin), because both could be regarded as being defined on the product of our two sample spaces (the set of pairs (die face, coin flip)).
