Ratio of CDF to PDF increasing? Let $\Phi(x)$ be a cumulative normal distribution function and $\phi(x)$ the associated probability density function.
Is the ratio $\frac{\Phi(x)}{\phi(x)}$ increasing in x?
Numerically it seems to be true. Is there any ways to prove it analytically?
Thanks

As in Conrado Costa's answer:
$$
\Bigl(\frac{\Phi(x)}{\phi(x)}\Bigr)' = 1 + x \frac{\Phi(x)}{\phi(x)} \tag{1}
$$
and the RHS is clearly positive for $x\ge0$.
For $x<0$ write
$$
1 + x \frac{\Phi(x)}{\phi(x)}
  =  \frac{\phi(x)}{x} \, \Psi(x) ,
$$
where
$$
\Psi(x) := \frac{\phi(x)}{x} + \Phi(x) .
$$
$\Psi(x)$ is negative for $x<0$
since $\lim_{x\to-\infty}\Psi(x) = 0$ and
$\Psi'(x) := -\frac{\phi(x)}{x^2} < 0$,
hence the RHS of (1) is positive for $x<0$ too.
