TABLE OF CONTENTS 3.1. X and Y be independent normal variables with parameters ( ? 1, ? 1) and ( ? 2, ? 2). Using formulas (3.150) and (3.157), derive the MLEs and the UMVUEs of R when ? 1 and ? 2 are known and ? 1 and ? 2 are unknown, or when ? 1 and ? 2 are known and ? 1 and ? 2 are unknown.
3.2. Let X and Y be independent lognormal variables with parameters ( ? 1, ? 1) and ( ? 2, ? 2), where the pdf of the lognormal distribution is given in (3.42). Using monotone transformation v(x)=ln x, formulas (3.150) and (3.157) and Theorems 2.7-2.9 derive the MLEs and the UMVUEs of R when ? 1 and ? 2 are known and ? 1 and ? 2 are unknown, or when ? 1 and ? 2 are known and ? 1 and ? 2 are unknown, or all parameters are unknown.
3.3. Let X~Beta( ? 1, ? 1) and Y~Beta( ? 2, ? 2) be independent beta-variables where the pdf of the beta distribution is defined in (3.11). Derive the MLE (3.41) of R=P(X< Y) and show that the series is convergent.
3.4. Let X and Y be independent...
