/sci/ - Science & Math » Thread #12775601

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Anonymous Tue 02 Mar 13:57:16 2021 No.12775601 View Reply Original Report

Quoted By: >>12775683 >>12776578 >>12776998

How do I solve this? More than a hint would be appreciated. Completely lost.

Anonymous

Anonymous Tue 02 Mar 2021 14:30:12 No.12775683 Report

Quoted By: >>12776374

>>12775601
just write out what happens when you go through the cases when mu < c, Y < c and so on

Anonymous

Anonymous Tue 02 Mar 2021 17:57:47 No.12776374 Report

Quoted By:

>>12775683
how do you understand

Anonymous

Anonymous Tue 02 Mar 2021 18:51:38 No.12776578 Report

Quoted By:

>>12775601
Let f(x) = max(x,c)
Clearly f(x) is convex (just graph it).
Since f is convex, you can use jensen's inequality.
https://en.wikipedia.org/wiki/Jensen%27s_inequality#Measure-theoretic_and_probabilistic_form
E[max(Y,c)] = E[f(Y)] = integral[f(y)*p(y)dy] >= f(integral[y*p(y)dy]) = max(E[Y],c).

Anonymous

Anonymous Tue 02 Mar 2021 20:19:21 No.12776998 Report

Quoted By:

>>12775601
Just write $\mathrm{Max}(Y,c) = Y 1_{Y \geq c} + c 1_{Y < c}$ and conclude that both $\mathbb{E}[\mathrm{Max}(Y,c)] \geq \mathbb{E}(Y)$ and $\mathbb{E}[\mathrm{Max}(Y,c)] \geq c$ .

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