Let me assume that the variable $x\in [a,b]$ and $y\in[a,b]$ where $a,b (a<b)$ are real numbers. I have positive increasing function $g:[a,b] \rightarrow [c,d]$, where $c,d(c<d)$ are real numbers. $$g(x') \ge g(x) \text{ and } g(x)\ge 0$$ for all $x'\ge x$.

And I have a pdf $f:[a,b]\times [a,b] \rightarrow [0,1]$. More specifically, $$f(x,y) = \frac{\frac{1}{\sqrt{2\pi \sigma^2}}exp(-\frac{(x-y)^2}{2\sigma^2})}{\int_{a}^{b}\frac{1}{\sqrt{2\pi \sigma^2}}exp(-\frac{(x-y)^2}{2\sigma^2})dx},$$ which means the boosted $y$-mean Gaussian distribution for $x$ on the interval $[a,b]$.

I want to prove, for fixed $x$, $$\int_{a}^{b}g(x)f(x,y)dx$$ is increasing for $y$ or not. I think it should be increasing, but don't know how to prove this. Actually, I want to find a pdf which makes the last integral form increase by using Normal distribution.


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