A special case of a bivariate non-central t-distribution

Let X1 and X2 have a bivariate normal distribution with means zero and variances one, and correlation p. Let Y be distributed independent of the X's and let Y have a square root of a chi-square (with f degrees of freedom) divided by f-distribution, where the square root extends over the f in the denominator also. Then (Xl + 8)/ Y and (X2 + 82)/Y have non-central t-distributions with f degrees of freedom and non-centrality parameters 81 and 62, respectively. A bivariate non-central t-distribution may then be defined as the joint distribution of (X + 81)/ Y and (X2 +82)/ Y. This definition is in conformance with the definition of a multivariate t-distribution given by Dunnett & Sobel (1954). In this paper, we will be interested in the special case of probabilities of joint events associated with the pair of random variables 1Tf = (X + 81)/ Y and 2Tf = (X + 82)/ Y, i.e. where X1 = X2 = X (or p = 1). Certain of these probabilities have applications to two-sided tolerance limits and two-sided sampling plans. The connexion between these two-sided tolerance limits, twosided sampling plans and one-sided sampling plans and tolerance limits will also be discussed. We will also use the notation Tf = (X + d)! Y for a random variable with the univariate noncentral t-distribution.