$X$ is a non negative random variable with decreasing density function. Let $U$ be a $Unif(0,2t)$ random variable where $t>0$. For $x>0$ define $G(X)=P(X>x)$. Then show that $$\mathbb{E}(G(U)) \geq G(\mathbb{E}(U))$$
So, basically $G(X)=1-F(X)$, where $F(X)$ is the distribution function of $X$. Now, $G'(X)=-f(x)$ , $f(x)$ being the density function of $X$. And, $G''(x)=-f'(x)$ which is positive as $f$ is decreasing. This gives us $G$ to be a convex function and then I apply Jensen inequality to get the proof. But there is a problem. In the question it's not said whether $f$ is differentible or not. So, I have to prove first, that $f$ is differentiable and then I can do the rest.
I don't know how to show this.
Any help or any different method would be highly appreciated!
Thanks!
Best Answer
You don't require differentiability. The indefinite integral of any integrable decreasing function is concave, so $F=1-G$ is concave and $G$ is convex.
PS: There is link below given by Minus One-Twelfth which has a neat proof of convexity.