The Distribution of the Identified Minimum of a Normal Pair Determines' the Distribution of the Pair

Let (Xo , X1) have a bivariate normal density with mean (,u0 , 1.) and covariance matrix (-i,) where a,i = ai2 and o-i = po-,o (i, j = 0, 1 i # j). Let Z = min (Xo, X1) and define I by Z = XI . The situation wherein (Xo, X1) is not observable but where we may observe (Z, I) arises in reliability problems and elsewhere (e.g. one may model the lifetime Z of some object in this way when death may occur as a result of one of two causes which are observable via autopsy). The object of this note is to establish the following Proposition: The distribution H of the (observable) pair (Z, I) uniquely determines the distribution of the (unobservable) pair (Xo , X1). We need the following Lemma: Let n(. a, b2) denote the univariate normal density with mean a and standard deviation b, let N(. I a, b2) be the corresponding probability integral. Then the conditional density of Z given I = i, for all z and i = 0, 1, is given by f (z) P'n (z I i , a2)(1 N((z ,*)/oT* I 0, 1) if pal-i F ( [n(z 2 L, , ao-) otherwise where