Let say $X \sim N \left(0, 1\right)$
and, $Y = \Phi\left( \frac{a-bX}{c} \right), a,b,c, >0$, which $\Phi$ is the cdf of Standant Normal distribution.
I wish to calculate the ${\alpha}^{th}$ quantile of $Y$.
Can it be said that the ${\alpha}^{th}$ quantile of $Y$ is $\Phi\left( \frac{a-b X_{{\alpha}^{th}}}{c} \right)$, where $X_{{\alpha}^{th}}$ is the ${\alpha}^{th}$ quantile of the random variable $X$?
Any pointer will be very helpful
Let $y_{\alpha}$ be the $\alpha-$quantile of $Y$. By the definition of the quantile,
$$\mathbb P\left[Y > y_\alpha\right] = \alpha\quad \implies \quad\mathbb P\left[X < \frac{a - c\Phi^{-1}\left(y_\alpha\right)}{b}\right] = \alpha \quad \implies \quad \mathbb P\left[X > \frac{a - c\Phi^{-1}\left(y_\alpha\right)}{b}\right] = 1-\alpha$$ So again by the definition of the quantile, $$\frac{a - c\Phi^{-1}\left(y_\alpha\right)}{b} = x_{1-\alpha}$$ finally $$y_\alpha = \Phi\left(\frac{a - b x_{1-\alpha}}{c}\right)$$