Which of the following statements are correct?

(A) The mean and variance of the Poisson distribution are equal.
(B) The mean and variance of a Binomial distribution are equal
(C) An unbiased die is thrown again and again until two sixes are obtained, then the probability of obtaining the second six in the 3rd throw is $\frac{5}{108}$.
(D) If the variance of a Poisson distribution is 2, then $P(X = 2) = 2e^{-2}$

Choose the correct answer from the options given below:

(A), (C) and (D) only

The correct answer is Option (1) → (A), (C) and (D) only **

(A) True. For Poisson distribution, mean = variance = $\lambda$.

(B) False. For Binomial distribution, mean $=np$ and variance $=np(1-p)$ (equal only if $p=\frac12$).

(C) True. Second six on 3rd throw means: First two throws → exactly one six, Third throw → six.

Probability = $\frac{1}{6}\cdot\frac{5}{6}\cdot\frac{1}{6} + \frac{5}{6}\cdot\frac{1}{6}\cdot\frac{1}{6} = \frac{10}{216}=\frac{5}{108}$.

This matches the given number, so (C) is True.

(D) Variance = $\lambda = 2$. $P(X=2)=\frac{2^2 e^{-2}}{2!}=2e^{-2}$. True.

Correct statements: (A), (C), (D)