In zero-modified Pn, E[N] Modified equals which expression?

• A. ((1 - Pm0) / (1 - p0)) * E[N]
• B. E[N] * (1 - p0) / (1 - Pm0)
• C. E[N] / (1 - p0)
• D. E[N] * (1 - Pm0)

Answer: (B)

Zero-modified means you change only the probability of zero and renormalize the nonzero part so the total probability is 1, leaving the shape of the positive counts proportional to the original. If P(N=0) = p0 and P(N=n) = pn for n ≥ 1, with sum_{n≥1} pn = 1 − p0, and you set the zero probability to Pm0, the nonzero probabilities must be Pm(n) = c · pn for n ≥ 1 where c is chosen so that sum_{n≥1} Pm(n) = 1 − Pm0. This gives c = (1 − Pm0)/(1 − p0). The modified expectation becomes E[N Modified] = sum_{n≥1} n · Pm(n) = c · sum_{n≥1} n · pn = [(1 − Pm0)/(1 − p0)] · E[N]. Therefore the modified expectation is E[N] multiplied by (1 − Pm0)/(1 − p0).