Given a normal random variable , how can we approximate it by a discrete distribution with only two realizations  and  ?

To begin with, given the symmetry of the normal distribution, a natural choice is to take two equiprobable realizations, , symmetric with respect to the expected value. To find the displacement , we may match variance, which is equivalent to matching the second order moment:

which boils down to . Note that, by symmetry, we also match skewness, i.e., the third-order central moment. We leave it as an exercise to prove that if we add a third point , we find . With five points, we may also afford to match the fourth-order moment, kurtosis, which is 3 for any normal variable.

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X~ N(,2)