This paper demonstrates how averaging over individual learning and forgetting curves gives rise to transformed averaged curves. In an earlier paper (Murre and Chessa, 2011), we already showed that averaging over exponential functions tends to give a power function. The present paper expands on the analyses with exponential functions. Also, it is shown that averaging over power functions tends to give a log power function. Moreover, a general proof is given how averaging over logarithmic functions retains that shape in a specific manner. The analyses assume that the learning rate has a specific statistical distribution, such as a beta, gamma, uniform, or half-normal distribution. Shifting these distributions to the right, so that there are no low learning rates (censoring), is analyzed as well and some general results are given. Finally, geometric averaging is analyzed, and its limits are discussed in remedying averaging artefacts.

本文演示了对个体学习和遗忘曲线进行平均如何产生转换后的平均曲线。在早期的论文（Murre 和 Chessa，2011）中，我们已经证明对指数函数求平均往往会给出幂函数。本文扩展了指数函数的分析。此外，结果表明，对幂函数进行平均往往会给出对数幂函数。此外，还给出了对对数函数的平均如何以特定方式保留该形状的一般证明。分析假设学习率具有特定的统计分布，例如 beta、gamma、均匀或半正态分布。还分析了将这些分布向右移动，以便不存在低学习率（审查），并给出一些一般结果。最后，