### Chapter 7, Survey Sampling

#### Solution 25

The joint distribution of any subcollection of is same as of :

Since each permutation has an equal probability, the joint distribution of the subcollections must be identically distributed i.e. the joint distribution of every subcollection is same. Thus the joint probability for any subcollection is same.

Now we can check easily is equal to . For simplicity, consider for the case : is the probability that first and second item of a random permutaion of are and respectively. This is clearly the same as drawing two items from items such that the first item drawn is and second item drawn is . This in turn is same as . Similarly the result holds true for any size .

Now, it is easy to see . Since and because and are identically distributed, the required result follows.

Similarly, since the joint distribution of any subcollection of the permutations is identically distributed as that of a simple random sample, it follows that .

Now, since , or a constant, it follows that variance of

Using the variance formulae,

Thus we get

$$\tag*{$\blacksquare$} $$