why the expectation of X is equal to $$sum(x.p(x))$$ , or the integral in the case of the continuous variable. I mean where this definition cames from?
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$\begingroup$Here is how I understand it intuitively. Recall that the probability of an event $E$ is the proportion of all the experiments in which $E$ happens. Now if you conduct many many experiments, each outcome $x$ is expected to occur with proportion $p(x)$, so the total outcome after $N$ experiments is expected to be$$\sum x(N\cdot p(x))$$where the sum is ranging of all possible $x$. Then the expected value is defined to be the average of this sum, giving you the expected value of the outcome of each experiment.
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