# About a class of expectations

Consider being given a $$n-$$dimensional random vector with a distribution $${\cal D}$$, vectors $$a \in \mathbb{R}^k$$, $$\{ b_i \in \mathbb{R}^n \}_{i=1}^k$$ and non-linear Lipschitz functions, $$f_1,f_2 : \mathbb{R} \rightarrow \mathbb{R}$$.

Now under what conditions can we exactly compute (or give tight bounds on)

$$\mathbb{E} \Big [ x f_1 \Big ( \sum_{i=1}^k a_i f_2 (b_i^\top x) \Big ) \Big ]$$

as a function of the the $$a$$ and $$b_i$$ vectors ?

I can hardly find any examples of non-trivial $$f_1$$ and $$f_2$$ (and $${\cal D}$$ being anything "reasonable" like log-concave etc.) for which this is computable!

• For an exact and non-trivial example, try a standard bivariate normal distribution with the function $f(x)=\max(-x,x/2)$. – Matt F. Mar 14 at 8:38
• Could you kindly elaborate? I am actually quite surprised that this is the $f$ you chose. I had in mind initially a very similar $f$! – gradstudent Mar 15 at 1:16
• Eg one part of that expectation could be wolframalpha.com/input/… – Matt F. Mar 15 at 3:47
• Sorry! Could you kindly say some more details? Like the expression I have written above is a vector. But you are computing a scalar. Could you kindly check that? And what are your $a_i$s and the $b_i$s? – gradstudent Mar 15 at 4:58
• That’s part of why I said “one part”. I don’t think these are research-level questions. – Matt F. Mar 15 at 10:50