Expected Distinct Units Calculation
The problem asks for the expected number of distinct units drawn when sampling 2 units from a population of 20 with replacement, using probability proportional to size (PPS) sampling. The probabilities $p_i$ are given.
Formula for Expected Distinct Units
For sampling with replacement, the expected number of distinct units ($E(V)$) in a sample of size $n$ from a population of size $N$ is given by:
$E(V) = N - \sum_{i=1}^{N} (1 - p_i)^n$
Here, $N = 20$ and the sample size $n = 2$.
Given Probabilities
- For units $i = 1, \cdots, 10$ (10 units): $p_i = \frac{1}{40}$.
- For units $i = 11, \cdots, 20$ (10 units): $p_i = \frac{3}{40}$.
Let's verify the sum of probabilities: $10 \times \frac{1}{40} + 10 \times \frac{3}{40} = \frac{10}{40} + \frac{30}{40} = \frac{40}{40} = 1$. The probabilities are correctly normed.
Calculation Steps
We need to calculate the term $\sum_{i=1}^{N} (1 - p_i)^n$. Since $n=2$, we calculate $(1 - p_i)^2$.
- Calculate $(1 - p_i)^2$ for the first group of 10 units:
For $p_i = \frac{1}{40}$, $(1 - p_i)^2 = \left(1 - \frac{1}{40}\right)^2 = \left(\frac{39}{40}\right)^2 = \frac{1521}{1600}$. - Calculate $(1 - p_i)^2$ for the second group of 10 units:
For $p_i = \frac{3}{40}$, $(1 - p_i)^2 = \left(1 - \frac{3}{40}\right)^2 = \left(\frac{37}{40}\right)^2 = \frac{1369}{1600}$. - Calculate the sum $\sum_{i=1}^{20} (1 - p_i)^2$:
The sum consists of 10 terms of the first type and 10 terms of the second type.
$ \sum_{i=1}^{20} (1 - p_i)^2 = 10 \times \left(\frac{39}{40}\right)^2 + 10 \times \left(\frac{37}{40}\right)^2 $ $ = 10 \times \frac{1521}{1600} + 10 \times \frac{1369}{1600} $ $ = \frac{15210}{1600} + \frac{13690}{1600} = \frac{28900}{1600} = \frac{289}{16} $ - Calculate the Expected Number of Distinct Units $E(V)$:
$ E(V) = N - \sum_{i=1}^{20} (1 - p_i)^2 $ $ E(V) = 20 - \frac{289}{16} $ $ E(V) = \frac{20 \times 16}{16} - \frac{289}{16} = \frac{320}{16} - \frac{289}{16} = \frac{320 - 289}{16} $ $ E(V) = \frac{31}{16} $
Final Answer
The expected number of distinct units drawn is $\frac{31}{16}$.
Important Questions from Ratio And Regression
- Consider a finite population of size $N$. Let $T_1$ be the sample mean based on a sample of size $n$ under simple random sampling with replacement (SRSWR) scheme. Let $T_2$ be the sample mean based on a stratified random sample of size $n$ where the samples are drawn from each of 4 strata using SRSWR scheme under proportional allocation. Then which of the following are sufficient conditions for $\text{Var}(T_1) = \text{Var}(T_2)$ to hold?
Suppose there are $k$ strata of $N = kM$ units each with size $M$. Draw a sample of size $n_i$ with replacement from the $i^{\text{th}}$ stratum and denote by $\bar{y}_i$ the sample mean of the study variable selected in the $i^{\text{th}}$ stratum, $i = 1, 2, \dots, k$. Define
$$ \bar{y}_s = \frac{1}{k}\sum_{i=1}^k \bar{y}_i \text{ and } \bar{y}_w = \frac{\sum_{i=1}^k n_i \bar{y}_i}{n} $$
Which of the following is necessarily true?- Suppose we draw a random sample of size $n$ from a population of size $N$, where $1 < n < N$, using simple random sampling without replacement scheme. Let $P$ be the population proportion of units possessing a particular attribute and $p$ be the corresponding sample proportion. Which of the following is an unbiased estimator for $P(1 - P)$?
Suppose there are $k$ groups each consisting of $N$ boys. We want to estimate the mean age $\mu$ of these $kN$ boys. Fix $1 < n < N$ and consider the following two sampling schemes.
I. Draw a simple random sample without replacement of size $kn$ out of all $kN$ boys.
II. From each of the $k$ groups draw a simple random sample with replacement of size $n$.
Let $\bar{Y}$ and $\bar{Y}_G$ be the respective sample mean ages for the two schemes. Which of the following are true?
- A sample of size $n (\ge 2)$ is drawn without replacement from a finite population of size $N$, using an arbitrary sampling scheme. Let $\pi_i$ denote the inclusion probability of the $i$-th unit and $\pi_{ij}$, the joint inclusion probability of units $i$ and $j, 1 \le i < j \le N$. Which of the following statements is always true?
