<Tr> <Th> MGF </Th> <Td> (1 − p 1 − p e t) r for t <− log ⁡ p (\ displaystyle (\ biggl () (\ frac (1 - p) (1 - pe ^ (t))) (\ biggr)) ^ (\! r) (\ text (for)) t <- \ log p) </Td> </Tr> <Tr> <Th> CF </Th> <Td> (1 − p 1 − p e i t) r with t ∈ R (\ displaystyle (\ biggl () (\ frac (1 - p) (1 - pe ^ (i \, t))) (\ biggr)) ^ (\! r) (\ text (with)) t \ in \ mathbb (R)) </Td> </Tr> <Tr> <Th> PGF </Th> <Td> (1 − p 1 − p z) r for z <1 p (\ displaystyle (\ biggl () (\ frac (1 - p) (1 - pz)) (\ biggr)) ^ (\! r) (\ text (for)) z <(\ frac (1) (p))) </Td> </Tr> <Tr> <Th> Fisher information </Th> <Td> r (1 − p) 2 (p) (\ displaystyle (\ frac (r) ((1 - p) ^ (2) (p)))) </Td> </Tr>

Write the equation for p demand = 0 as a function of r and p in the negative binomial model