# mean and standard deviation from probability distribution function

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] - Âµ^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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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 ...

#### Solution Summary

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