The subject treated in this thesis is the conditional distribution of a random variable given that the outcome of an associated random variable lies within a specified interval. This may be considered to be an extension of the classical case in which the outcome of the associated random variable is known to assume a specific numerical value.

The primary purpose of the study was to examine the properties of a system formed by interval conditioning under the assumption of a suitable linear model. No attention was given to appropriate estimation procedures.

The principal conclusions of the study follow. Let X and Y be jointly distributed random variables such that E(Y|X) = α + βX, where α and β are constants, and such that the variance of Y given X is independent of X. Then E(Y|X∈ I) = α + β E(X|X∈ I) and the variance of Y given X∈ I is equal to the variance of Y given X plus β² times the variance of X in its truncated distribution, i.e. truncated in the conditioning interval I.

It was shown that the limiting cases of the system. led to the classical conditional results as the conditioning interval degenerates to a point, and to the classical marginal results as the interval expands to encompass the real line. These results were generalized into the case where a random variable Y is conditioned on a set of associated variables, {X_i}^p_i=1, such that X_i∈ I_i, i = 1, 2, … p.

Higher conditional moments were found in general. Since third and higher conditional moments are usually functions of the conditioned variables, only an analytic form was given.

Consideration was given to the case in which a vector of random variables is to be predicted given that an associated vector of random variables lies in a specified rectangular region. Two types of conditioning were considered simultaneously at this point, namely, the case in which part of the associated variables are conditioned to points and the remainder to intervals.

In various places in the body of the thesis and in the appendix consideration was given to the conditions under which the variance of a truncated random variable increases monotonically with the interval of truncation. This was found to be a complicated problem, but necessary and sufficient conditions for this property were developed in the appendix.