A sample of size n is selected from the population of N units. Another sample of same size is taken from remaining N-n units. If y'1 (y prime 1)and y'2 are the means of 1st and 2nd sample and Y¯ (Y bar) is the mean of then show that
E(y¯1/ y¯2) = (Y¯ - ny¯1) / N-n and cov (y¯1, y¯2) = -s2/N
I just need a start on how the steps will lead. . how do we calculate it?
Dear Hamna,
This is a simple exercise of conditional expectation. For it, it is necessary to understand the concept of conditional expectation which is accessible from the concept of mathematical expectation. There are some mistakes in your formulae (of the formulated Question). For example, in the first expectation it would be:
E(y-2|y-1) = E[(NY--ny-1)/(N-n)|y-1] = (NY--ny-1)/(N-n)
And
Cov(y-1, y-2) = E[Cov(y-1, y-2)|y-1] + Cov[E(y-1|y-1), E(y-2|y-1)]
But they are correct. Could you follow the reasoning?
Dear Hamna,
If you understand Spanish, this book would be of your interest:
Ruiz Espejo, Mariano (2013). Exactitud de la Inferencia en Poblaciones Finitas. Bubok. Madrid.
http://www.bubok.es/libros/229317/EXACTITUD-DE-LA-INFERENCIA-EN-POBLACIONES-FINITAS
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