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Practice Problem: Recover the probability mass function from the characteristic function
A discrete random variables X has a moment generating (characteristic) function $ M_X(s) $ such that
$ \ M_X(j\omega)= 3+\cos(3\omega)+ 5\sin(2\omega). $
Find the probability mass function (PMF) of X.
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Answer 1
I tried taking the inverse fourier transform since PX(x) = F^-1 { Mx(jw)}, however my resultant pmf has j (sqrt(-1)) in the answer, and doesn't sum to 1...
Are we finding an invalid pmf or am i approaching the problem wrong?
-AW
- It is much easier to expand the moment generating function in terms of complex exponentials and then compare the resulting summation with the formula for the moment generating function to find the values of the PMF. -pm
Answer 2
Write it here.
Answer 3
Write it here.
