Which of the following statements about the Poisson distribution are correct?
(A) The mean and the variance of a Poisson distribution are equal.
(B) The Poisson distribution is a limiting case of the binomial distribution when the number of trials n is very large, the probability of success p is very small and np remains finite.
(C) If X follows a Poisson distribution with mean m, then \(\displaystyle\frac{P(X = r + 1)}{P(X = r)} = \frac{m}{r + 1}\) for r = 0, 1, 2, ...
(D) A Poisson variate can take only the values 0, 1, 2, ..., n, where n is a fixed number of trials.
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
A Poisson variate X with mean m takes the values r = 0, 1, 2, 3, ... (with no upper limit), and
\( \displaystyle P(X = r) = \frac{e^{-m}\,m^{r}}{r!} \)
Dividing two neighbouring probabilities:
\( \displaystyle \frac{P(X = r + 1)}{P(X = r)} = \frac{e^{-m}\,m^{r+1}}{(r + 1)!} \times \frac{r!}{e^{-m}\,m^{r}} \)
\( \displaystyle = \frac{m}{r + 1} \)
So the mean and variance are both m, the distribution arises from the binomial when n is large, p is small and np = m is finite, and X has no upper limit.
Checking the statements
(A) The mean and the variance of a Poisson distribution are equal. — Correct. Both are equal to m.
(B) The Poisson distribution is a limiting case of the binomial distribution when n is very large, p is very small and np remains finite. — Correct. This is the limiting case of the binomial, with m = np.
(C) \(\frac{P(X = r + 1)}{P(X = r)} = \frac{m}{r + 1}\) — Correct. The factor \(e^{-m}\) cancels and \(\frac{m^{r+1}}{m^{r}} \times \frac{r!}{(r+1)!} = \frac{m}{r + 1}\).
(D) A Poisson variate can take only the values 0, 1, 2, ..., n. — Incorrect. It can take every value 0, 1, 2, 3, ... with no upper limit; a fixed n belongs to the binomial distribution.