We study the randomized $n$-th minimal errors (and hence the complexity) of vector valued mean computation, which is the discrete version of parametric integration. The results of the present paper form the basis for the complexity analysis of parametric integration in Sobolev spaces, which will be presented in Part 2. Altogether this extends previous results of Heinrich and Sindambiwe (J.\ Complexity, 15 (1999), 317--341) and Wiegand (Shaker Verlag, 2006). Moreover, a basic problem of Information-Based Complexity on the power of adaption for linear problems in the randomized setting is solved.

翻译：本文研究向量值均值计算的随机$n$阶最小误差（即复杂度），该问题是参数化积分的离散版本。本文结果为Sobolev空间中参数化积分的复杂度分析奠定了基础，相关分析将在第二部分中呈现。整体而言，这扩展了Heinrich与Sindambiwe（J.\ Complexity, 15 (1999), 317--341）以及Wiegand（Shaker Verlag, 2006）的先前结果。此外，本文还解决了基于信息复杂度理论中一个基本问题——随机设定下线性问题的自适应性作用。