4. If X is an integer-valued random variable with characteristic function ø, show that 1 P(X = k) = - 27 -itk o (1) dt. What is the corresponding result for a random variable whose distribution is arithmetic with span 2 (that is, there is probability one that X is a multiple of 2, and 1 is the largest positive number with this property)?

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A:

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A: Solution by using Gram-Schmidt process as follows :

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A: We can use the concept of Pythagoras theorem to solve the given problem.