TABLE OF CONTENTS In this section we shall obtain UMVUEs for a number of basic univariate distributions. Here we shall assume that X and Y are independent. See Section 3.5 for the case of dependent normal variables. The estimators are based on the samples X =( X 1, X 2, , 
), and Y =( Y 1, Y 2, , 
) (see (2.2)).
Let, as in Section 3.1.1, X and Y be independent normal variables with means ? 1, ? 2 and variances 
and 
, respectively. As it follows from Theorem 2.3, the UMVUE of R=P(X
and 
which are defined in (2.14) and (3.3), respectively, if all of the parameters of the underlying distributions are unknown. A number of authors constructed the UMVUE of R for this popular case, among them Mazumdar (1970), Downton (1973), Chandra and Owen (1975), Woodward and Kelley (1977), Voinov (1984), Mukherjee and Sharan (1985) and Rukhin (1986).
Initially, all effort has been devoted to the case when X has known parameters ? 1 and ? 1 (Mazumdar (1970), Downton (1973), Woodward and Kelley (1977)). The UMVUE of the pdf of a normal distribution with unknown mean and variance based on n 2 observations Y is of the form
| (3.49) |  |
Therefore, it follows from (2.28) that
