We typically have multiple data observations which we use to estimate a paramete

Question

We typically have multiple data observations which we use to estimate a parameter of interest. (For example, in estimating the height of the average student, we would collect a random sample of students and measure their heights, and then use the sample mean as an estimation of the population mean.) We nearly always assume that our observations are drawn ‘iid’—independent and identically distributed. Why is the assumption that the observations are independent mathematically convenient? Think about the product term in the likelihood function, and how this simplifies conditional probabilities if the observations are independent.

Explanation / Answer

As explained in the problem, even if we draw the units from different schools, the assumption of independence means that the values, say math scored, of one student does not influence that of another student. If we take the observations to be drawn from one school, it would be fair to assume that the observations are independent as well as identical. Here, the students are more or less from same socio-economic class, they learn from at the same time as well as from the same teacher. Thus, the assumptions of identical-ness is also fair here. However, if we are comparing the scores of students from a rural place to another urban one, the assumption of independent holds good while that identical not so.

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