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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?

St.artistics
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1 Answers1

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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.

Ray
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  • Great explanation, I just have one question about the $N$ in the $$\sum x(N\cdot p(x))$$ is really inside the sommatory or it is in the index of the $$\sum$$ ? Thank you for the answer – St.artistics Oct 09 '20 at 03:38
  • The index of the sum would be the set of all possible $x$. I use $N$ to stand for the total number of experiments (i.e. a fixed number). I choose to not write it just for simplicity. Formally speaking, this inexing set is called the sample space of the experiment. – Ray Oct 09 '20 at 03:43
  • thank you for the answer – St.artistics Oct 09 '20 at 18:56