Assume that we have $$P(\alpha) = Q(\sqrt{\alpha \rho})$$ where $Q(\alpha)$ is the Gaussian Q function defined as $$ Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty \exp\{-\frac{u^2}{2}\} ~du$$
Can anyone kindly let me know the steps for averaging the Q function over a chi square distribution. In other words, to solve the following integral
$$P_{avg} = \int_0^\infty Q(\sqrt{\alpha \rho}) ~ \frac{1}{(2n-1)!} ~ \alpha^{2n-1} ~ e^{-\alpha} d\alpha$$
Appreciate your help.