Answer by p4sch for Is the function $e^{x^2/2} \Phi(x)$ monotone increasing?
A useful inequality is\begin{equation}\tag{1}\frac{1}{\sqrt{2 \pi}} \frac{x}{x^2+1} \mathrm{e}^{-x^2/2} \le \frac{1}{\sqrt{2\pi}} \int_x^\infty \mathrm{e}^{-y^2/2} \, \mathrm{d} y \le \frac{1}{\sqrt{2...
View ArticleAnswer by cardinal for Is the function $e^{x^2/2} \Phi(x)$ monotone increasing?
This is just an alternative argument to Davide's nice one.First, note that $h' = e^{x^2/2}(x \Phi + \Phi')$.Since $\Phi'' = -x \Phi'$, monotonicity of the integral yields$$x \Phi(x) \geq...
View ArticleAnswer by Igor Rivin for Is the function $e^{x^2/2} \Phi(x)$ monotone...
I haven't actually done the computation, but it seems to me that integrating the $\Phi(x)$ term by parts ad nauseam, you get a nice power series for $h(x).$EDIT @Davide's argument is obviously the...
View ArticleAnswer by Davide Giraudo for Is the function $e^{x^2/2} \Phi(x)$ monotone...
We can write $h(x)=\frac 1{\sqrt{2\pi}}\int_{-\infty}^x \exp\left(\frac{x^2-y^2}2\right)dy$. Now put $t=x-y$. We get \begin{align}h(x)&=\frac...
View ArticleIs the function $e^{x^2/2} \Phi(x)$ monotone increasing?
Hello,Here is an interesting problem. It looks elementary, but it has taken me some efforts without solving it. Let$$h(x) = e^{x^2/2} \Phi(x),\qquad \text{with}\quad \Phi(x):=\int_{-\infty}^x...
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