Approximate Solutions for Means and Variances in a Certain Class of Box Problems

Abstract

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.