Random Sample From an Infinite Universe

It is relatively difficult to explain the concept of random sample from an infinite population. However, a few examples will show the basic characteristics of such a sample. Suppose we consider the 10 throws of a fair dice as a sample from the hypothetically infinite population that consists of the results of all possible throws of the dice. If the probability of getting a particular number, say 7, is the same for each throw and the 10 throws are all independent, then we say that the sample is random. Similarly, it would be said to be sampling from an infinite population if we sample with replacement from an infinite population and our sample would be considered as a random sample if in each draw all elements of the population have the same probability of being selected and successive draws happen to be independent. In brief, one can say that the selection of each item in a random sample from an infinite population is controlled by the same probabilities and that successive selections are independent of one another.

In other words, if we have to take a sample of grain from a bag, it is not possible to assign a number to each grain or particle constituting the universe and as such the methods of constructing card population or of random sampling numbers cannot be used. In such cases a thorough mixing of the grain may be done and by dividing and sub-dividing the lot in parts, a sample of an adequate size can be obtained. The contents of the bag after thorough mixing may be divided in two equal parts of which one may be selected and this may further be divided in two parts after mixing. In this way the process can be continued till one of the sub-divisions is equal to the size of the desired sample.

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