ON THE EFFECT OF TRUNCATION IN SOME OR ALL COORDINATES OF A MULTINORMAL POPULATION
WASHINGTON UNIV SEATTLE LAB OF STATISTICAL RESEARCH
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A p-dimensional normal random variable X sub-1, X sub-2,..., X sub-p may represent p quantitative traits of an individual very often an admission test requires that each of these traits be above a certain preassigned value, so that only those individuals pass the test for whom X sub-1 t sub1,0000, X sub p t sub-p. Explicit expressions are obtained for the moments EX sub-i, EX sub-i squared, EX sub-i X sub-j, and the procedure for extending the method to the general case of EX sub-i to the power of m X sub-j to the power of n is indicated. The possibility of truncation in some but not all coordinates is included since, for example, the case of X sub-1 not truncated, X sub-2 truncated at tau corresponds to t sub-1 -infinity, t sub-2 tau. Explicit expressions are also obtained for the marginal probability density function of X sub-1 and for the joint marginal probability density function of X sub-1, X sub-2, after truncation in X sub1, X sub2,...., X sub-p. Examples are given for the use of some of the results for determining t sub-1, t sub-2,......so that certain preassigned changes in the population are achieved.
- Statistics and Probability