I have to resolve the follow problem about basic sampling concepts. I've tried a solution, I want to know if I've solved the problem well, please
In planning an office network study, the following draw sequential sampling scheme was proposed for selecting a randome sample of two nonadjacent office hour intervals from the eight intervals 9-10, 10-11,...16-17 (labeled 1, 2, 3...8):
- Draw the first hour interval for the sample with equal probability from the eight intervals
- Draw, without replacement, the seconid hour interval with equal probability from the intervals not adjacent in time to the one selected in the first draw.
a. Determine the first-order inclusion probabilities
For me we have to divide in two cases:
- If $k = 1$ or $8$ and $k \in \{2,3,...7\}$
At first case I define: $$ \pi_{k}= \sum{p(s)} = \frac{6}{21} $$ 'cause the probability of a single sample is $\frac{1}{21}$ and there is $6$ cases to choose $k =1$ or $k=8$
In the case that $k \in \{2, 3, ...7 \}$ then:
$$ \pi_{k}= \sum{p(s)} = \frac{5}{21}$$ 'cause there is $5$ cases to choose.
b. Determine the second-order inclusion probabilities. Is the design induced by the proposed sampling scheme measurable?
Here if we consider that any sample of this exercise will have only two elements, then the second-order probability would be:
$$ \pi_{k,l} = \sum{p(s)} = \frac{1}{21} $$
Because for every pair of elements $k$ and $l$ we have only one posibility to select him.
c. Verify the following propierties
a. $ \sum_{U}{\pi_{k}} = n$
I have (a) $ \displaystyle \pi_{k} = \frac{5}{21}$ at $6$ times (2,3,...,7), then is equal to $\displaystyle \sum{\pi_{k}}=\frac{30}{21}$ On the other hand, I have $\displaystyle \pi_{k}=\frac{6}{21}$ at $2$ times (1 or 8) then is equal to $ \displaystyle \sum{\pi_{k}}= \frac{12}{21}$ And finally $\displaystyle \sum_{U}{\pi_{k}} = \frac{30}{21} + \frac{12}{21} = 2$ as wanted (n = 2 is the sample size).
Finally I have to prove that $$ \sum\sum_{k \neq l}{\pi_{k,l}} = n(n-1)$$ and $$\sum_{ \displaystyle k \neq l, l\in U}{\pi_{k,l}= (n-1)\pi_{k}} $$ But I don't know how to correctly check these last two properties.
I appreciate any help