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Given a random variable X, define the moment generating function
if X and Y are independent, how does the MGFX+Y(•) simplify?
MGFX(•) • MGFY(•)
similarly E(et(x+y)) = E(etx) • E(ety)
obtain the mgf by differentiating
what is the power/taylor series defined as?
define the probability generating function
what is the formula for turning the PGF into a probability formula?
This process proves that the PGF and the probability distribution contain the exact same information. If PX(t) is identical to PY(t), then decoding them will produce the exact same set of probabilities. Therefore, X and Y must be identically distributed.