• Expectation of Geometric Distribution:
• Let $X$ denote the number of independent tosses required until the first head appears in tossing a fair coin.
• The probability of getting a head on any single toss is $p = 0.5$, and tails is $q = 1 - p = 0.5$.
• The random variable $X$ follows a Geometric distribution:

• The expected value (mean number of tosses) for a geometric distribution is:

• Conclusion: The expectation of the number of tosses required is 2 (Option 3).

Correct option: (3) 2

• Standard Error of Estimated Proportion:
• Sample size: $n = 36$
• Number of accounts with deposit < Rs. 1500: $x = 12$
• Sample proportion: $p = \frac{x}{n} = \frac{12}{36} = \frac{1}{3}$, and $q = 1 - p = \frac{2}{3}$
• In simple random sampling from a very large population, the unbiased estimator of the variance of the sample proportion is:

• The estimated standard error ($SE$) is the square root of the variance:

• Conclusion: The estimated standard error is $\sqrt{\frac{2}{315}}$ (Option 4).

Correct option: (4) on

• Comparison of Variances in SRSWR and SRSWOR:
• Let $V_1$ be the variance of the sample mean based on all $n$ units under SRSWR:

• Let $V_2$ be the variance of the sample mean based on distinct units $d$ under SRSWR (using the Rao-Blackwellized estimator $\bar{y}_d$, where $E[d] \lt n$):

• Let $V_3$ be the variance of the sample mean under SRSWOR of the same fixed size $n$:

• Theoretical Ordering: Since repeated sampling adds redundant units in SRSWR, discarding duplicate units strictly improves efficiency ($V_1 \ge V_2$). Furthermore, a direct SRSWOR sample of fixed size $n$ always achieves at least as many distinct units as SRSWR, giving:

• Conclusion: The relationship that holds is $V_1 \ge V_2 \ge V_3$ (Option 1).

Correct option: (1) V42V22V3

• Estimation of Population Total in Simple Random Sampling:
• Population size: $N = 120$ households
• Sample size: $n = 4$ households
• Household sizes in sample: $y = \{6, 4, 7, 3\}$
• Sample mean persons per household:

• The unbiased estimator of the population total $\hat{Y}$ is:

• Conclusion: The estimate of the total number of people in the village is 600 (Option 3).

Correct option: (3) 600