It is shown that certain monomials in normally distributed quantities have stable distributions with index $2^{-k}$. This provides, for $k > 1$, simple examples where the mean of a sample has a distribution equivalent to that of a fixed, arbitrarily large multiple of a single observation. These examples include distributions symmetrical about zero, and positive distributions. Using these examples, it is shown that any distribution with a very long tail (of average order $\geq x^{-3/2}$) has the distributions of its sample means grow flatter and flatter as the sample size increases. Thus the sample mean provides less information than a single value. Stronger results are proved for still longer tails.

Using the notation of Levene and Wolfowitz [1], a new recursion formula is used to give the exact distribution of arrangements of $n$ numbers, no two alike, with runs up or down of length $p$ or more. These are tabled for $n$ and $p$ through $n = 14$. An exact solution is given for $p \geq n/2$. The average and variance determined by Levene and Wolfowitz are presented in a simplified form. The fraction of arrangements of $n$ numbers with runs of length $p$ or more are presented for the exact distributions, for the limiting Poisson Exponential, and for an extrapolation from the exact distributions. Agreement among the tables is discussed.

Let $F(P)$ be a real valued function defined on a subset $\mathscr{D}$ of the set $\mathscr{D}^\ast$ of all probability distributions on the real line. A function $f$ of $n$ real variables is an unbiased estimate of $F$ if for every system, $X_1, \cdots, X_n$, of independent random variables with the common distribution $P$, the expectation of $f(X_1 \cdots, X_n)$ exists and equals $F(P)$, for all $P$ in $\mathscr{D}$. A necessary and sufficient condition for the existence of an unbiased estimate is given (Theorem 1), and the way in which this condition applies to the moments of a distribution is described (Theorem 2). Under the assumptions that this condition is satisfied and that $\mathscr{D}$ contains all purely discontinuous distributions it is shown that there is a unique symmetric unbiased estimate (Theorem 3); the most general (non symmetric) unbiased estimates are described (Theorem 4); and it is proved that among them the symmetric one is best in the sense of having the least variance (Theorem 5). Thus the classical estimates of the mean and the variance are justified from a new point of view, and also, from the theory, computable estimates of all higher moments are easily derived. It is interesting to note that for $n$ greater than 3 neither the sample $n$th moment about the sample mean nor any constant multiple thereof is an unbiased estimate of the $n$th moment about the mean. Attention is called to a paradoxical situation arising in estimating such non linear functions as the square of the first moment.

In this paper significance tests are developed whose application requires only the determination of one order statistic and the computation of sums of sample values. The simplest case considered is that of testing a new sample value $x$ on the basis of $m$ previous sample values $y_1, \cdots, y_m,$ all sample values being assumed from normal populations with the same variance. Two separate tests of whether the mean of the new population from which $x$ was taken exceeds the mean of the population from which $y_1, \cdots, y_m$ were drawn consist in accepting the alternative that the new population mean exceeds the old population mean if \begin{equation*}\tag{1} x > \big(\frac{\sqrt{m + 1} + 1}{m}\big)\sum^m_1 y_i - \sqrt{m + 1} y_{(u)}\end{equation*} \begin{equation*}\tag{2} x > \big(\frac{\sqrt{m + 1} - 1}{m}\big)\sum^m_1 y_i + \sqrt {m + 1} y_{(m+1-u)},\end{equation*} where $y_{(u)}$ is the $u$th largest of $y_1, \cdots, y_m$. It can be shown that both of these tests have the same power so that either one might be equally well selected for use. In practical application, however, there may exist reasons for preferring one test to the other. Similarly, the alternative that the new population mean is less than the old population mean will be accepted if \begin{equation*}\tag{3} x < \big( \frac{\sqrt{m+1} + 1}{m}\big) \sum^m_1 y_i - \sqrt{m + 1} y_{(m+1-u)}\end{equation*} \begin{equation*}\tag{4} x < \big(\frac{\sqrt {m + 1} - 1}{m}\big)\sum^m_1 y_i + \sqrt{m + 1} y_{(u)}.\end{equation*} All four of these significance tests have the same power, also the same significance level $\alpha (u, m)$. By appropriate choice of $u$ and $m$ the significance level can be made to assume values suitable for significance tests. For example, $\alpha (1, 6) = .0156,\quad\alpha (2, 10) = .0107$ $\alpha (3, 13) = .0110,\quad \alpha (4, 16) = .0107.$ The above tests are still valid if each of $x, y_1, \cdots, y_m$ equals a sum of $r$ sample values. These order statistic tests are generalized to the case where $x$ is a sum of $r$ new sample values; $y_1, \cdots, y_m$ each equals a sum of $s$ past sample values and another sum of relatively weighted past sample values is utilized but not as an order statistic. The introduction of this relatively weighted sum allows less reliable past information to be lumped together and weighted according to its relative importance. In comparing the order statistic tests with the most powerful tests which could be used for these alternatives it is found that the size of the samples used must be increased in order to bring the efficiency of the order statistic test up to that of the corresponding most powerful test. Thus the advisability of using the order statistic test will depend upon whether it is more desirable to take larger samples but have less computation.

The negative binomial distribution of Greenwood and Yule is generalized and modified in order to obtain distribution curves which could be used in many concrete cases of chains of rare events. Assuming that the numbers of single, double, triple, and so on, events are distributed according to Poisson's law with parameters $\lambda_1, \lambda_2, \lambda_3 \cdots$ respectively, and that $\lambda_s$ is given by $\lambda_s = \lambda_1 \frac{a^{s - 1}}{s!}$, the probability of obtaining $M$ successful events is studied. In the considered relation $\lambda_s$, for convenient values of $a$, first increases with $s$ and after a certain saturation value of $s$ starts to decrease. A relation of this type is very suitable for studying the distribution of score in a match between two first class billiard players, the probability of accidents on a highway of dense traffic, etc. The general methods of finding the distribution curves for arbitrary relations between the $\lambda$'s are indicated. The method of steepest descent is applied to find an acceptable approximation of the distribution function; and the advantage of this method is pointed out for other similar cases, in addition to the concrete one which was developed, in which the method of direct expansion into power series becomes inapplicable.