How to calculate $\int\limits_{-\infty}^{+\infty} \frac{n-1}{\alpha}\Phi(\frac{x}{\alpha})^{n-2}\phi(\frac{x}{\alpha})^2dx$?

Question

I was working on a research project that involves taking the integral of

$$\frac{n-1}{\alpha}\int\limits_{-\infty}^{+\infty} \Phi\left(\frac{x}{\alpha}\right)^{n-2}\phi\left(\frac{x}{\alpha}\right)^2dx,$$ where $\alpha>0$ and $n\in\mathbb{Z}$ and $n>3$.

Eventually, I wish to show that the whole expression decreases monotonically as $\alpha$ increases.

Any help on this will be greatly appreciated. Thanks!

Answer 1

As clearly shown by Guy in his comment, the substitution $x=\alpha u$ leads to $$\int\limits_{-\infty}^{+\infty} \frac{n-1}{\alpha}\Phi\Big(\frac{x}{\alpha}\Big)^{n-2}\phi\Big(\frac{x}{\alpha}\Big)^2dx=(n-1)\int\limits_{-\infty}^{+\infty} \Phi(u)^{n-2}\phi(u)^2du$$ which is independant of $\alpha$.

If you need to go further, could you clarify, at least for me, what are functions $\Phi(u)$ and $\phi(u)$ ?