Under which of the following conditions the Poisson distribution is the limiting case of binomial distribution:
(A) The number of trials is indefinitely large.
(B) The probability of success for each trial is indefinitely small.
(C) The product of the number of trials and the probability of success for each trial is finite.
(D) The probability of success for each trial is indefinitely large.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (C) only
The Poisson distribution is the limit of the binomial distribution when \(n \to \infty\), \(p \to 0\) and \(np = \lambda\) stays finite. Then, for each k,
\( \displaystyle {}^{n}C_{k}\,p^{k}(1 - p)^{n - k} \to \frac{e^{-\lambda}\lambda^{k}}{k!} \)
(A) The number of trials is indefinitely large. — Correct. This is \(n \to \infty\).
(B) The probability of success for each trial is indefinitely small. — Correct. This is \(p \to 0\).
(C) The product of the number of trials and the probability of success for each trial is finite. — Correct. This is \(np = \lambda\), the mean of the Poisson distribution.
(D) The probability of success for each trial is indefinitely large. — Incorrect. The probability of success must be very small; it cannot also be large.