Randomized Complexity of Parametric Integration and the Role of Adaption I. Finite Dimensional Case

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.