I am trying to do some practice questions with respect to properties of the gaussian distribution:
$\int_{-\infty}^{\infty} e^{-\frac{1}{2}(ax^2 + \beta x + \gamma)}N(\delta x + \epsilon)dx$
where $N(y) = \frac{1}{ \sqrt{2 \pi}} \int_{-\infty}^{y}e^{-x^{2}/2}$. I am provided with a hint to differentiate by $\epsilon$
I would greatly appreciate it if someone could help me.
Thanks
Following the suggestion, differentiate this expression with respect to $\epsilon$ to get
$$ \int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}}\exp\left(-\tfrac{1}{2}(ax^{2}+bx+c+(\delta x+\epsilon)^{2})\right)\mathrm{d}x =\frac{\exp\left(-\frac{1}{2}c-\frac{1}{2}\epsilon^2+\frac{(b+2\delta\epsilon)^{2}}{8a+8\delta^{2}}\right)}{\sqrt{a+\delta^{2}}} $$
Integrate with respect to $\epsilon$ and use either of the values at $\epsilon\rightarrow\pm\infty$. You will be able to write the result in terms of the $N$ function you defined.