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safina
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Homework Statement
A sample of size n[tex]_{1}[/tex] is to be drawn from a normal population with mean [tex]\mu_{1}[/tex] and variance [tex]\sigma^{2}_{1}[/tex]. A second sample of size n[tex]_{2}[/tex] is to be drawn from a normal population with mean [tex]\mu_{2}[/tex] and variance [tex]\sigma^{2}_{2}[/tex]. What is the maximum likelihood estimator of [tex]\theta[/tex] = [tex]\mu_{1}[/tex] - [tex]\mu_{2}[/tex]?
If we assume that the total sample size n = n[tex]_{1}[/tex] + n[tex]_{2}[/tex] is fixed, how should the n observations be divided between the two populations in order to minimize the variance of the maximum likelihood estimator of [tex]\theta[/tex]?