The guesswork refers to the distribution of the minimum number of trials needed to guess a realization of a random variable accurately. In this study, a non-trivial generalization of the guesswork called guessing cost (also referred to as cost of guessing) is introduced, and an optimal strategy for finding the $\rho$-th moment of guessing cost is provided for a random variable defined on a finite set whereby each choice is associated with a positive finite cost value (unit cost corresponds to the original guesswork). Moreover, we drive asymptotically tight upper and lower bounds on the logarithm of guessing cost moments. Similar to previous studies on the guesswork, established bounds on the moments of guessing cost quantify the accumulated cost of guesses required for correctly identifying the unknown choice and are expressed in terms of R\'enyi's entropy. Moreover, new random variables are introduced to establish connections between the guessing cost and the guesswork, leading to induced strategies. Establishing this implicit connection helped us obtain improved bounds for the non-asymptotic region. As a consequence, we establish the guessing cost exponent in terms of R\'enyi entropy rate on the moments of the guessing cost using the optimal strategy by considering a sequence of independent random variables with different cost distributions. Finally, with slight modifications to the original problem, these results are shown to be applicable for bounding the overall repair bandwidth for distributed data storage systems backed up by base stations and protected by bipartite graph codes.

翻译：猜测工作指准确猜出随机变量实现所需最小试验次数的分布。本研究引入猜测工作的一类非平凡推广——猜测代价（亦称猜测成本），针对定义在有限集上的随机变量提供了求解猜测代价ρ阶矩的最优策略，其中每个选择关联正有限代价（单位代价对应原始猜测工作）。进一步，我们推导出猜测代价矩对数的新近紧上下界。与先前关于猜测工作的研究类似，已建立的猜测代价矩界限量化了正确识别未知选择所需的累积猜测代价，并以Rényi熵形式表达。此外，我们引入新随机变量建立猜测代价与猜测工作之间的关联，从而导出衍生策略。建立这一隐含关联有助于获得非渐近区域的改进界限。据此，通过考虑具有不同代价分布的独立随机变量序列，我们利用最优策略建立了以Rényi熵率表示的猜测代价矩的猜测代价指数。最后，对原始问题进行适度修改后，这些结果可适用于受限基站支撑且受二分图码保护的分布式数据存储系统的总修复带宽界限。