Analytical properties of certain probability distributions - Kibble, W. F. (The University of Edinburgh, 1938)
This research has been carried out on Dr Aitken's suggestion. The first chapter is largely a recapitulation of known results which I have learnt from Dr Aitken, here arranged for convenience of reference later in the thesis.

The second chapter is the application of these methods to the deduction of a two-variate GamMa type Distribution. Dr Aitken pointed out that the variances in a normally correlated two-variate distribution would give the required distribution,and chanter II is just the carrying out of that suggestion. fie also directed me to the papers by Hardy and by Wishart and Bartlett which give rise to chapter III. Those two chapters form the centre or core of the thesis from which the other research radiates in three main directions, of varying interest from the points of view of pure mathematics and of statistical applications.

The first is chapter IV which is purely of mathematical interest. It contains the most substantial single piece of research in the thesis. It was perhaps fortunate that on my first searching Watson's Theory of Bessel Functions for theorems involving incomplete Gamma functions, I did not find the paragraph on Hadamadd's paper. The result given by Watson would have been sufficient for the purposes of chapter III, but not being satisfied with my own deduction of it, I sought to find it as a special case of a more general theorem, and in doing so have been led to discover a more general result, and incidental results which may be of great interest in themselves.

Secondly, the generalisations in chapters V and VI are of interest as giving forms for statistical distributions, but they wilt also be of interest for pure mathematics, giving, for example; a generalisation to any number of variables of Mehler s theorem (1866) which has been discussed by Hardy and Watson and others.

The third branch of the thesis is,concerned with the actual method of fitting a distribution function such as that discussed in chapters II and III. This is discussed in Chapter IX, and tables to make easy the fitting by the method suggested there of a type III curve are given in chapter X.

Other parts of the thesis are chapters VII and VIII, of which. VII is concerned with an attempt, not so far successful, to extend the theory to all Pearson's types, instead of type III only. (Similar attempts must have been made before s see Romanovsky, Biometrika, Vol XVI, parts I and II, p. 106) And chapter VIII applies a method whit I learnt from Dr Aitken in application to a normal distribution to the distribution of chapter II, and deduces simple formal which may be of great practical significance in dealing with problems of selection in distributions in which the coefficient of variation is important and is not small