# Standard Normal Distribution-Inequality

Suppose that $$\Phi(.)$$ and $$\phi(.)$$ are the cdf and the pdf of the standard Normal distribution, respectively. Does the following inequality hold $$\forall x>0$$: $$2x\phi(x)+2\Phi(x)-1\geq 0$$

• Indeed it is, but i had to find a partial derivative which was about 3 pages in order to make it in an appropriate form and this was the only easy part (that i missed after all :P), that I was not sure for the sign...Now i can say that the whole derivative is positive and that's all! :) – Nav89 Aug 2 at 9:45

For $$x>0$$ we have $$\Phi(x)>\frac12$$, hence $$2\Phi(x)-1 \geq 0$$ and since $$x\phi(x)$$ is positive for positive $$x$$ it clearly holds.