Consider $n$ boxes, each box having an associated probability, $p_i, \big(\sigma_i p_i = 1\big)$, and an associated integer, $k_i$. If balls are thrown one by one into these boxes, the probability being $p_i$ that any one ball falls into the $i$th box, then the number of balls which must be thrown in order to obtain, for the first time, at least $k_{i_1}$ balls in the $i_1$th box, at least $k_{i_2}$ balls in the $i_2$th box, $\cdots$, and at least $k_{i_s}$ balls in the $i_s$th box, is a random variable, $N_s\lbrack k_1(p_1), k_2(p_2), \cdots, k_n(p_n)\rbrack$. Here $i_1, i_2, \cdots, i_s$ represent the numbers of that set of $s$ boxes, $(1 \leq s \leq n)$, which first satisfies the stated condition. The distribution of $N_s\lbrack k_1(p_1), k_2(p_2), \cdots, k_n(p_n)\rbrack$ can be written down for any set of values assigned to $n, s$, the $p_i$'s and the $k_i$'s. However, for $n$ greater than 2 the distribution assumes such an extremely complicated multinomial form that except for certain special cases even the mean of the distribution cannot be numerically evaluated without a prohibitive amount of labor. This paper presents the exact moments of $N_1\lbrack k_1(p_1), k_2(p_2)\rbrack$ and $N_2\lbrack k_1(p_1), k_2(p_2)\rbrack$ in forms that readily lend themselves to computation and shows how these moments can be used to obtain approximate values for the mean and variance for certain situations where $n$ is greater than two. These approximation formulae are given for 1. The mean and variance, for any $n$ and any set of $k_i$'s and $p_i$'s when $s = 1$ or $n$. 2. The mean, for any $n$ and $2 \leq s \leq n - 1$, when $p_i = 1/n, k_i = k, (i = 1, 2, \cdots, n)$. Some indications are given concerning the error of the approximations, and the circumstances which lead to a minimum (and maximum) error. Curves have been prepared to show the mean for the two box case, the primary function of these curves being to assist in the application of the approximation formulae. Some problems where the results of this paper might be applicable are suggested in the Introduction.

The asymptotic distribution of the range $w$ for a large sample taken from an initial unlimited distribution possessing all moments is obtained by the convolution of the asymptotic distribution of the two extremes. Let $\alpha$ and $u$ be the parameters of the distribution of the extremes for a symmetrical variate, and let $R = \alpha(w - 2u)$ be the reduced range. Then its asymptotic probability $\Psi(R)$ and its asymptotic distribution $\psi(R)$ may be expressed by the Hankel function of order one and zero. A table is given in the text. The asymptotic distribution $g(w)$ of the range proper is obtained from $\psi(R)$ by the usual linear transformation. The initial distribution and the sample size influence the position and the shape of the distribution of the range in the same way as they influence the distribution of the largest value. If we take the parameters from the calculated means and standard deviations, the asymptotic distribution of the range gives a good fit to the calculated distributions for normal samples from size 6 onward. Consequently the distribution of the range for normal samples of any size larger than 6 may be obtained from the asymptotic distribution of the reduced range. The asymptotic probabilities and the asymptotic distributions of the mth range and of the range for asymmetrical distributions are obtained by the same method and lead to integrals which may be evaluated by numerical methods.

The means, variances, and covariances for samples of size $\leq 10$ from the normal distribution, a selected long-tailed distribution, and the uniform distribution are tabled and compared with the usual asymptotic approximations. The methods of computation used and the accuracy expected are discussed. Use is made of the representation of an arbitrarily distributed variate as a monotone function of a uniformly (rectangularly) distributed variate. It is hoped that these tables will encourage experimentation with new statistical procedures.

We consider sequential procedures for obtaining confidence intervals of prescribed length and confidence coefficient for the mean of a normal distribution with known variance. A procedure achieving these aims is called optimum if it minimizes the least upper bound (with respect to the mean) of the expected number of observations. The result proved is that the usual nonsequential procedure is optimum.