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Now my book defines another function mgf(x) i.e. moment generating function of x which is defined as: -

mgf(x) = E(e

^{tx})

I don't understand why was this function defined. Basically we included e

^{tx}in our function because then if we differentiate it, we get the E(x) formula back again at t = 0. Why take such pain?

I can only see one use where p(x) is such that mgf(x) forms a nice taylor series of some function which can be easily differentiated but other than that I see no use.