# mean and standard deviation from probability distribution function

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For simple probability distributions such as that corresponding to throwing an unbiased dice, it is easy just to calculate the mean and standard deviation from the standard formulae: mean = E[X] = ∑x*p(x) and σ^2 = E[X^2] - µ^2

However for more complicated distributions, it is usually neater to use the moment generating function E[Exp[tX]] or the characteristic function E[Exp[itX]]

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The expert examines mean and standard deviation from probability distribution function.

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We can use either E[Exp[tX]]or E[Exp[itX]], however the second one is preferable as we can always be sure it converges.

We know ∑p(x) = 1 by definition, so Modulus[E[Exp[itX] ]=Modulus[∑Exp[itX]*p(x) ] which is less than or equal to ∑Mod[Exp[itX]]*p(x) which is less than or ...

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