• Properties of Probability Distributions:
• Statement (A): Mean and variance are equal for Poisson distribution:
• In a Poisson distribution with parameter $\lambda$, $E(X) = \lambda$ and $\text{Var}(X) = \lambda$. Therefore, Mean = Variance. (Correct)
• Statement (B): Mean is greater than variance for binomial distribution:
• For $X \sim \text{Bin}(n, p)$, $\text{Mean} = np$ and $\text{Var}(X) = npq = np(1 - p)$. Since $0 \lt p \lt 1$, $q \lt 1$, hence $np \gt npq$. (Correct)
• Statement (C): Mean is less than variance for Chi-square distribution:
• For $\chi^2$ distribution with $k$ degrees of freedom, $\text{Mean} = k$ and $\text{Var} = 2k$. For any $k \ge 1$, $k \lt 2k$, meaning mean is strictly less than variance. (Correct)
• Conclusion: Statements (A), (B), and (C) are all correct (Option 4).

Correct option: (4) (A), (B) and (C)