• Ratio of Two Independent Standard Normal Variates:
• Let $X, Y \sim \text{N}(0, 1)$ be independent standard normal random variables.
• Let $Z = \frac{Y}{X}$. Using the transformation of variables technique:

• Evaluating this integral yields:

• This is the standard Cauchy distribution (with location parameter 0 and scale parameter 1), which is also identical to Student's $t$-distribution with 1 degree of freedom ($t_1$).
• Conclusion: The distribution of $Y/X$ is Standard Cauchy (Option 3).

Correct option: (3) Standard Cauchy

• Expected Number of Binomial Outcomes:
• Number of tosses per set: $n = 10$
• Probability of head on a single toss: $p = 0.5$, tails: $q = 0.5$
• Probability of getting exactly 7 heads and 3 tails in a set of 10 tosses:

• Total number of sets conducted: $N = 100$
• The expected number of sets showing 7 heads is:

• Conclusion: 12 cases are expected (Option 1).

Correct option: (1) 12

• Negative Binomial Distribution Properties:
• Statement (I): In a negative binomial distribution, mean never exceeds variance. $\implies$ Correct.
• For $X \sim \text{NB}(r, p)$ (number of failures before the $r$-th success):

• Since $0 \lt p \lt 1$, $\frac{1}{p} \gt 1$, which mathematically guarantees $\text{Variance} \gt \text{Mean}$. Thus, the mean never exceeds variance (overdispersion property).
• Statement (II): Negative binomial distribution is a generalised form of geometric distribution. $\implies$ Correct.
• The geometric distribution is the special case of the negative binomial distribution where the required number of successes is $r = 1$.
• Conclusion: Both Statement (I) and Statement (II) are correct (Option 1).

Correct option: (1) Both Statement (I) and Statement (I) are correct