[simplotek]

> If you want to introduce continuous distributions like the Gaussian one, you can just say "area under the curve" if you need to connect the density to a numerical probability.

What name do you give to this "area under the curve", or the "rate of change" of this area? They are pretty fundamental concepts with important and basic properties, which affect things like local optima and minimization, and expected value and covariance, etc. I mean, you can't cover linear models and least squares without this stuff, and if you don't then I wouldn't really call it learning.

[comte7092]

You call “area under the curve”… area under the curve. Expected values, least squares, linear model, etc can all be explained in the discrete case without calculus.

High school math isn’t and doesn’t need to be rigorously proofed based, if you lack some do the tooling necessary to demonstrate a proof, you can tell a student, “the proof requires calculus” and boom, you’ve given them a reason to take an interest in the subject.

[mturmon]

You don’t need integration to define expected value or covariance in the discrete case. TBH I’m not sure if you can get around integration in the general continuous case or not.

If not, you could use some limiting argument to handle the moments of a continuous uniform RV, at least, in terms of the discrete analog.

You don’t need calculus to derive least squares estimators. You can follow the logic in this quora answer [1] to show that (e.g.) the mean is the minimum MSE estimator among constant functions, and that the conditional mean is the minimum MSE estimator among “general” (measurable L2) functions.

This derivation is familiar to many who have studied these concepts. It’s clever, it does not need differentiation, just expectation and logic.

It could be that your studies in probability were done using a certain pedagogical path, and that’s blinding you to the fact that other paths are possible.

[1] https://www.quora.com/Why-is-minimum-mean-square-error-estim...