Question

Can someone explain the expected value formula in probability to me? Like why does it work? I have searched for proofs and tried to do the proof on my own, doesn't work

Answer 1

the expected value is a definition. There is nothing to prove.

Answer 2

if you are looking for motivation for the definition. (looking at the discrete case)

\[E(X)=\sum_{x\in\Omega}xP(X=x)\]

think of this as x is the position of a particle on a mass-less rod and P(X=x) is the mass of the particle. One can then show that the balancing point of the rod is the expected value.