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We study the randomized $n$-th minimal errors (and hence the complexity) ofvector valued mean computation, which is the discrete version of parametricintegration. The results of the present paper form the basis for the complexityanalysis of parametric integration in Sobolev spaces, which will be presentedin 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 ofadaption for linear problems in the randomized setting is solved.

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