Let X11 X12X1, and X21, X22X2nz be two independent random samples of size ni and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively. (a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first population (X₁). Using analogy, state the mle's of the parameters of the second population. (b) Find the pooled estimator of the common variance when it is assumed that of = o=0². Suggest an unbiased estimator of o². (c) Assuming both ny and n₂ are small, suggest a pivotal function that can be used to derive a (1-a) x 100% confidence interval for (₁-₂) when o² = 0₂ = 0².

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