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] - 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]]
© BrainMass Inc. brainmass.com March 4, 2021, 6:25 pm ad1c9bdddfhttps://brainmass.com/statistics/probability/mean-standard-deviation-from-probability-distribution-function-43753
Solution Preview
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.