Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). The function \(Q\) is strictly increasing on \((0,1)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For every \(a\in (0,1)\), \(Q\) is continuous at \(a\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For every \(a\in (0,1)\), \(Q'(a)=1/I(a)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For every \(a\in (0,1)\), \(I'(a)=-Q(a)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For every \(a\in (0,1)\), the derivative of \(p\mapsto -Q(p)\) at \(a\) is \(-1/I(a)\).

The standard Gaussian density satisfies \(\varphi (-x)=\varphi (x)\) for every \(x\in \mathbb R\).

For the standard Gaussian distribution function \(\Phi \), the function \(t\mapsto \Phi (-t)+\Phi (t)\) has derivative zero at every \(x\in \mathbb R\).

The standard Gaussian distribution function satisfies \(\Phi (-x)=1-\Phi (x)\) for every real \(x\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For every \(a\in (0,1)\), \(Q(1-a)=-Q(a)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For every \(a\in (0,1)\), \(I(1-a)=I(a)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(a,b\in (0,1)\), \(a\le b\), and \(a+b\le 1\), then \(Q(b)^2\le Q(a)^2\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(a,b\in (0,1)\), \(a\le b\), and \(a+b\le 1\), then \(I(a)\le I(b)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). The function \(p\mapsto Q(p)^2\) is convex on \((0,1)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For \(a\in (0,1)\), the derivative of \(p\mapsto I(p)Q(p)\) at \(a\) is \(1-Q(a)^2\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For \(a\in (0,1)\), the derivative of \(p\mapsto I(p)^2\) at \(a\) is \(-2I(a)Q(a)\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. If \(a\in (0,1)\), then \(\overline I(a)=\varphi (\Phi ^{-1}(a))\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Then \(\overline I(0)=0\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Then \(\overline I(1)=0\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. For every real \(a\), \(\overline I(a)\ge 0\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. For every real \(a\), \(\overline I(1-a)=\overline I(a)\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Then \(\lim _{p\to 0^+}\overline I(p)=0\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. The function \(\overline I\) is continuous at zero.

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. The function \(\overline I\) is continuous at one.

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. For \(a\in (0,1)\), \(\overline I'(a)=-\Phi ^{-1}(a)\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. The function \(\overline I\) is continuous on \([0,1]\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. The function \(\overline I\) is continuous on \(\mathbb R\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. If \(\mu \) is a finite measure on a measurable space \(\Omega \) and \(f:\Omega \to \mathbb R\) is measurable with \(f(x)\in [0,1]\) for every \(x\), then \(\overline I\circ f\) is \(\mu \)-integrable.