Feature engineering on distributions Something that has always bothered me is summarizing distributions when feature engineering for a machine learning process. Does anybody have best practices for this?

Example: Imagine a dataset showing orders of products by customers. You want to summarize customer behaviour.

In that example, if we're to only focus on customers' order value, the order value for each customer would have some distribution D(x).

Now, can create variables describing features of each customer's distributions (min, mean, median, max, quartiles, IQR, etc.), but are there best practices around which features tend to provide the most information upon extraction? Moreover, is there some way to contain the information of a distribution in a single variable?
At the end each distribution can be described by a function with parameters. Can be a Gaussian, polynomial etc. In principle you can choose functions that only have one free parameter and fit this one. Depending on your data you might be able to guess a function class that seems to fit the underlying distributions. You can then use the fit parameters as input for your data set. Example: Fit a Gaussian and use mean, normalization and sigma = 3 parameters for your model.