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Let $\pi^*_i \quad i=1,2,\dotsc, N$ be the probability of not including the $i$th unit in the sample and let $\pi^*_{ij} \quad i, j = 1, 2,\dotsc, N$ be the probability of including neither the $i$th nor the $j$th units in the sample. Then, show that

$$\pi^*_i \pi^*_j - \pi^*_{ij} = \pi_i \pi_j - \pi_{ij} $$

where is $\pi_i $ the probability of including the $i$th unit in the sample and $\pi_{ij} $ is the probability of including both the $i$th and the j th units.

Where $\ P_i$ is the probability of selection of $\ i^{th} $ unit from the population without replacement .

My solution $$\pi_i = P_i[1 + \sum_ {i=0}^N \frac {P_j}{1-P_j} ] $$

$$\pi_{ij} = \ P_i\ P_j [ \frac {1}{1-P_j} +\frac {1}{1-P_i} ] $$

Now if I work on the lhs of question by using

$$\pi^*_i = 1- P_i[1 + \sum_ {i=0}^N \frac {P_j}{1-P_j} - \frac{P_1}{1-P_1} ] $$

$$\pi^* _j = 1- P_j[1 + \sum_ {j=0}^N \frac {P_i}{1-P_i} - \frac{P_1}{1-P_1}] $$

$$\pi^*_{ij} = 1- \ P_i\ P_j [ \frac {1}{1-P_j} +\frac {1}{1-P_i} ]$$

But the problem is , if I am putting these values in left hand side then it is becoming a mess which looks nothing like rhs.

Putting everything together I am getting

$$- P_j[1 + \sum_ {j=0}^N \frac {P_i}{1-P_i} - \frac{P_1}{1-P_1}] - P_i[1 + \sum_ {i=0}^N \frac {P_j}{1-P_j} - \frac{P_1}{1-P_1} ] + \ P_iP_j[1 + \sum_ {j=0}^N \frac {P_i}{1-P_i} - \frac{P_1}{1-P_1}]\ ^2 + \ P_i\ P_j [ \frac {1}{1-P_j} +\frac {1}{1-P_i} ] $$

Need help in further procedure.

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  • $\begingroup$ What is $P_i$ in your solution? Also for sake of completeness you should mention that you are sampling without replacement from a population of size $N$. $\endgroup$
    – Dayne
    Commented Jul 2, 2021 at 7:02
  • $\begingroup$ @dayne I have done the respective changes, thanks for correcting. $\endgroup$
    – simran
    Commented Jul 2, 2021 at 15:31

1 Answer 1

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Draw a picture.

This is a Venn diagram of the two events $\mathscr{i}$ (unit $i$ is included in a sample) and $\mathscr{j}$ (unit $j$ is included in a sample). On it I have posted the inclusion probabilities, which I worked out by observing $\pi_i = \pi_{ij} + (\pi_i-\pi_{ij})$ and $\pi_j = \pi_{ij} + (\pi_j-\pi_{ij}).$

Figure

One axiom of probability tells us the chance of not including $i$ in the sample is $1$ minus the chance of including $i,$ since those chances must sum to $1.$ The same reasoning applies to $j,$ whence

$$\pi^{*}_i = 1-\pi_i\ \text{and}\ \pi^{*}_j = 1-\pi_j.$$

The chance that neither $i$ nor $j$ are included corresponds to all outcomes not enclosed in one of the circles. By the same reasoning, that chance ($\pi^{*}_{ij}$) must be equal to $1$ minus the chance of being in one of the circles. As the diagram shows, the latter is the sum of the three chances posted in the diagram (as asserted by another probability axiom: the chance of a disjoint union of events is the sum of the chances of the individual events). Therefore

$$\begin{aligned} \pi^{*}_{ij} &= 1 - \left[\left(\pi_1 - \pi_{ij}\right) + \pi_{ij} + \left(\pi_j - \pi_{ij}\right)\right] \\ &= (1-\pi_i)(1-\pi_j) - \pi_i\pi_j + \pi_{ij}\\ &= \pi^{*}_i\pi^{*}_j + \pi_{ij} - \pi_i\pi_j. \end{aligned}$$

Subtracting $\pi^{*}_i\pi^{*}_j$ from both sides yields the desired result.

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