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:

For a Poisson distribution with parameter $\lambda$:

Mean $= \lambda$

Variance $= \lambda$

Since both equal $\lambda$, the mean and variance are equal.

Statement (A) is correct.

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For a Binomial distribution with $n$ trials and probability $p$:

Mean $= np$

Variance $= np(1-p)$

The variance includes the factor $(1-p)$, which is less than 1, making the variance smaller than the mean.

For example, if $n = 10$ and $p = 0.5$:

Mean $= 10 \times 0.5 = 5$

Variance $= 10 \times 0.5 \times 0.5 = 2.5$

Statement (B) is incorrect.

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The second six occurs on the 3rd throw means:

• Exactly 1 six in the first 2 throws
• The 3rd throw is a six

Probability of exactly 1 six in first 2 throws:

$P = \binom{2}{1} \times \frac{1}{6} \times \frac{5}{6}$

$= 2 \times \frac{1}{6} \times \frac{5}{6}$

$= \frac{10}{36}$

Probability the 3rd throw is a six $= \frac{1}{6}$

Total probability:

$P = \frac{10}{36} \times \frac{1}{6}$

$= \frac{10}{216}$

$= \frac{5}{108}$

Statement (C) is correct.

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In a Poisson distribution, Variance $= \lambda$

If variance $= 2$, then $\lambda = 2$

The Poisson probability formula:

$P(X = k) = \frac{\lambda^k \times e^{-\lambda}}{k!}$

$P(X = 2) = \frac{2^2 \times e^{-2}}{2!}$

$= \frac{4 \times e^{-2}}{2}$

$= 2e^{-2}$

Statement (D) is correct.

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Statements (A), (C), and (D) are correct.