The importance of stochastic processes in relation to problems of population growth was pointed out by W. Feller [1] in 1939. He considered among other examples the "birth-and-death" process in which the expected birth and death rates (per head of population per unit of time) were constants, $\lambda_o$ and $\mu_o$, say. In this paper, I shall give the complete solution of the equations governing the generalised birth-and-death process in which the birth and death rates $\lambda(t)$ and $\mu(t)$ may be any specified functions of the time. The mathematical method employed starts from M. S. Bartlett's idea of replacing the differential-difference equations for the distribution of the population size by a partial differential equation for its generating function. For an account of this technique,$^1$ reference may be made to Bartlett's North Carolina lectures [2]. The formulae obtained lead to an expression for the probability of the ultimate extinction of the population, and to the necessary and sufficient condition for a birth-and-death process to be of "transient" type. For transient processes the distribution of the cumulative population is also considered, but here in general it is not found possible to do more than evaluate its mean and variance as functions of $t$, although a complete solution (including the determination of the asymptotic form of the distribution as $t$ tends to infinity) is obtained for the simple process in which the birth and death rates are independent of the time. It is shown that a birth-and-death process can be constructed to give an expected population size $\bar n_t$ which is any desired function of the time $t$, and among the many possible solutions the unique one is determined which makes the fluctuation, Var$(n_t)$, a minimum for all. The general theory is illustrated with reference of two examples. The first of these is the $(\lambda_0, \mu_1t)$ process introduced by N. Arley [3] in his study of the cascade showers associated with cosmic radiation; here the birth rate is constant and the death rate is a constant multiple of the "age, $t$, of the process. The $\bar n_t$-curve is then Gaussian in form, and the process is always of transient type. The second example is provided by the family of "periodic" processes, in which the birth and death rates are periodic functions of the time $t$. These appear well adapted to describe the response of population growth (or epidemic spread) to the influence of the seasons.

This paper contains a study of the following problem: Each of two events recurs with definitely known period and duration, while the starting time of each event is unknown. It is desired that, before the elapse of a certain time, the events occur simultaneously and that this "overlap" be of at least a given minimum duration. The probability of this satisfactory coincidence is first evaluated, and it is found that the solution, while mathematically adequate, is of no value for practical application. This circumstance arises from the possibility that, with certain rational ratios of the periods, the events may "lock in step". Accordingly, an attempt is made to smooth the probability function with respect to small variations in the ratio of the periods. Due to difficulties in manipulating the number-theoretic expressions involved, this smoothing is carried through only by the use of certain approximations. Moreover, because of these same difficulties, an averaged value of the probability itself is not obtained, but, in its stead, there is derived a formula for that fraction of randomly related repeated trials in which the original probability will be less than one-half. Thus, the original problem is not completely solved. The results obtained, however, do allow one to compare the relative advantages of different situations and to make a rough estimate of the likelihood of success. Generally speaking, the analysis is applicable whenever the ratio of "on time" to "off time" is small for each event.

In Paper II of this series [2, 1947] it was shown that if $n$ functions and a sample of $n$ were used to divide the population space into $n + 1$ blocks in a particular way, and if the joint cumulative of the functions were continuous, then the $n + 1$ fractions of the population, corresponding to the $n + 1$ blocks, were distributed symmetrically and simply. In Paper I of this series [1, 1945] it was shown that the one-dimensional theory of tolerance regions could be extended to the discontinuous case, if equalities were replaced by inequalities. In this paper the results of Paper II will be extended to the discontinuous case with the same weakening of the conclusion. The devices involved are more complex, but the nature of the results is the same (See Section 5). As a tool, it is shown that any $n$-variate distribution can be represented in terms of an $n$-variate distribution with a continuous joint cumulative (in fact, with uniform univariate marginals), where each variate of the given distribution is a different monotone function of the corresponding variate from the continuous distribution.

Asymptotic properties of maximum likelihood estimates have been studied so far mainly in the case of independent observations. In this paper the case of stochastically dependent observations is considered. It is shown that under certain restrictions on the joint probability distribution of the observations the maximum likelihood equation has at least one root which is a consistent estimate of the parameter $\theta$ to be estimated. Furthermore, any root of the maximum likelihood equation which is a consistent estimate of $\theta$ is shown to be asymptotically efficient. Since the maximum likelihood estimate is always a root of the maximum likelihood equation, consistency of the maximum likelihood estimate implies its asymptotic efficiency.

S. N. Roy [2] obtained in 1943 the distribution of the maximum, minimum and any intermediate one of the roots of certain determinantal equations based on covariance matrices of two samples on the null hypothesis of equal covariance matrices in the two populations. The present paper gives a different method of working out the distribution of any of these roots under the same hypothesis. The distribution of the largest, smallest and any intermediate root when the roots are specified by their position in a monotonic arrangement has been derived for $p = 2, 3, 4,$ and 5 by the new method. The method is applicable for obtaining the distribution of the roots of an equation of any order, when the distributions of the roots of lower order equations have been worked out.

A test is proposed for deciding whether one of $k$ populations has slipped to the right of the rest, under the null hypothesis that all populations are continuous and identical. The procedure is to pick the sample with the largest observation, and to count the number of observations $r$ in it which exceed all observations of all other samples. If all samples are of the same size $n, n$ large, the probability of getting $r$ or more such observations, when the null hypothesis is true, is about $k^{1-r}$. Some remarks are made about kinds of errors in testing hypothesies.

Conditions are determined for insuring that Neyman's method of constructing similar regions by means of sufficient statistics will yield all such regions when such statistics exist.

The greatest fractional increase in variance when a weighted mean is calculated with approximate weights is, quite closely, the square of the largest fractional error in an individual weight. The average increase will be about one-half this amount. The use of weights accurate to two significant figures, or even to the nearest number of the form: 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 55, 60, 65, 70, 75, 80, 85, 90, or 95, that is to say, of the form $10(1)20(20)50(5) 100 \times 10^r$ can thus reduce efficiency by at most $\frac{1}{4}$ percent, which is negligible in almost all applications.