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

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  • $\begingroup$ 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! :) $\endgroup$
    – Nav89
    Aug 2, 2019 at 9:45

1 Answer 1

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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.

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  • $\begingroup$ Thank you very much! $\endgroup$
    – Nav89
    Aug 2, 2019 at 9:42

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