I have a random variable $x \in [a, b]$ with PDF $f(x)$ and an event $E$ which satisfies the following property for any $x'<b$.


$$\Pr[E\mid x > x'] \geq \Pr[E]$$


My question is whether or not the following inequality holds.


$$\int\_a^b uf(u)\Pr[E\mid x=u] \, du \geq \Pr[E]\int\_a^b uf(u) \, du$$



Expected value of a random variable conditioned on a positively correlated event
The answer is yes. 
Indeed, let us write $X$ instead of $x$, according to standard notation, to distinguish between random variables (denoted by upper-case letters) and their values (denoted by lower-case letters). Let us write $P$ instead of $\Pr$, and then let us also write $A$ instead of $E$, to distinguish it from the expectation sign. 


Then we need to show that 
\begin{equation}
 EX\,P(A|X)\ge P(A)\,EX
\end{equation}
given that 
\begin{equation}
 P(A|X>y)\ge P(A)
\end{equation}
for all $y$ such that $P(X>y)\ne0$. 


Replacing $X$ by $X-a$, we may assume that $X\ge0$. Then 
\begin{multline\*}
 EX\,P(A|X)=EX\,E(1\_A|X)=EX\,1\_A=E\Big(\int\_0^X dy\Big)\,1\_A \\
 =E\Big(\int\_0^\infty dy\,1\_{X>y}\Big)\,1\_A
 =E\Big(\int\_0^\infty dy\,1\_{X>y}1\_A\Big) \\ 
 =E\int\_0^\infty dy\,1\_{X>y,A}=\int\_0^\infty dy\,E1\_{X>y,A} \\ 
 =\int\_0^\infty dy\,P(X>y,A) =\int\_0^\infty dy\,P(A|X>y)P(X>y) \\ 
 \ge\int\_0^\infty dy\,P(A)P(X>y)=P(A)\int\_0^\infty dy\,P(X>y)=P(A)EX, 
\end{multline\*}
as claimed. 


(The last equality follows immediately from the special case $EX=\int\_0^\infty dy\,P(X>y)$ of the previously proved equality $EX\,P(A|X)=\int\_0^\infty dy\,P(X>y,A)$, with $A$ replaced there by an event of probability $1$.) 


