Given some continuous probability distribution $\rho$ over $\mathbb{R}^3$-tuples $(a, w, b)$, define the function $$y(x) = \int a \max(0, wx+b) \rho(da,dw,db)$$

Can we construct a continuous probability distribution $\rho$ s.t. $y$ is a (non-constant) piecewise linear function over $\mathbb R$? Is that possible?

A couple things to note here:

1) For discrete or mixed discrete-continuous distributions this is fairly easy. Is that also possible for continuous distributions? By continuous I mean absolutely continuous w.r.t. the Lebesgue-measure, i.e. distributions whose support forms a non null-set w.r.t. the Lebesgue measure.

2) If one choses $a$ to be distributed symmetrically around $0$ (e.g. $a \sim \text{Unif}([-1,1])$), then $y$ is obviously equal to $0$ everywhere. Technically, this is a piecewise linear function, but a degenerate one. Can we construct a distribution such that $y$ is a non-constant piecewise linear function?

Any help is greatly appreciated! It would be nice to know whether this is possible and, if so, how. If you think that this isn't possible, any tips on a proof idea are also more than welcome! Thanks!

Edit: I am getting more and more convinced that this is not possible. I tried using the fact that piecewise linear functions have constant derivative on each interval and then differentiating the integral using Leibniz formula. So far, I haven't gained any major insight.


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