Suppose that I have $$\phi(-x)=1-c$$ where $\phi$ is the $N(0,1)$ cdf and $0\leq c \leq 1$. To find $x$ by applying the quantile function, do I then have $$-x = z_{1-c}\rightarrow x = -z_{1-c}$$
2026-03-25 20:40:19.1774471219
Quantile function for negative input
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Yes. The answer you provide is right.
From another perspective, by the property of $\phi(x)$, it can be shown $\phi(-x) = 1 - \phi(x)$, then $\phi(x) = c$ in your problem. Thus, $x = z_c = -z_{1-c}$.