The standard normal CDF has unique symmetric quantiles

Given: AC; the standard normal law $N(0,1)$; its CDF $\mathrm{\Phi }$; and a probability $q\in (0,1)$ when a quantile is considered.

The Axiom of Choice is assumed in the construction of the standard normal probability law. The Axiom of Choice Standard normal and normal laws

AC implies countable choice, which is the hypothesis needed for the zero-width-box measure result used to show singleton sets have zero normal probability. The Axiom of Choice implies countable choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The standard normal law is a probability measure with density $\varphi (x)=e^{-x^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }$; a probability measure is finitely additive on disjoint events. Standard normal and normal laws Probability measures and probability spaces

The density $\varphi$ is positive and Borel measurable, with integral one. The standard normal density has total mass one

The CDF of a real random variable is $\mathrm{\Phi }(x)=P(Z\le x)$. Cumulative distribution function of a real random variable

For increasing measurable sets, measure of the union is the limit of their measures. Continuity from below for measures

For decreasing measurable sets whose first member has finite measure, measure of the intersection is the limit of their measures. Continuity from above when one set has finite measure

A continuous function on a closed interval takes every value between its endpoint values. Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a,b]$ takes every value between $f(a)$ and $f(b)$

A monotone differentiable substitution with the stated local integrability hypotheses preserves a convergent improper integral, with orientation handled by the sign. Change of variable in an improper integral

A zero-width one-dimensional box has Lebesgue measure zero, under countable choice. A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included

The integral of a nonnegative measurable function over a measurable null set is zero. A nonnegative integral over a null set vanishes

The real exponential is smooth, so the displayed density $\varphi$ is continuous. The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$

If $A\subseteq B$, then $P(B\setminus A)=P(B)-P(A)$; in particular $P(A^c)=1-P(A)$ for a probability measure. Basic identities for a probability measure