Question: Find the moment-generating function, the characteristic function, and the first and second central moments of the following distributions: (a) Uniform distribution over the interval (a,b)

Find the moment-generating function, the characteristic function, and the first and second central moments of the following distributions:

(a) Uniform distribution over the interval (a,b)

(b) Gamma distribution with parameters X, r

(c) Geometric distribution

(d) Negative binomial distribution with probability mass function

Px(x) = r+x-1) - P )p(1 pr', - x = 0,1,... 0

Px(x) = r+x-1) - P )p(1 pr', - x = 0,1,... 0 otherwise.

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