The conditional expectation of X, given that X is less than x, is
-phi(x)/Phi(x). This conditional expectation must be less than x.
On 11/28/2012 10:17 PM, Esteban Gonzalez wrote:
> Probably I have not been clear enough about my question. phi(x) refers
> to the normal density function (dnorm in R) and Phi(x) to its
> corresponding distribution function (pnorm in R).
> It holds that phi(x)' = -x*phi(x) and therefore U'(x) = dnorm(x) > 0.
> Thus U(x) is a strictly
> increasing function.
> I need to demonstrate that U(x) = dnorm(x) + x*pnorm(x) > 0 for all x.
> Given that U(x) is strictly increasing, one way should be to check lim
> (x->-infinity) U(x) = 0 (something I haven't been able to work out).
> But probably it is not the only way to demonstrate my claim.
> Thank you in advance,
> Esteban
>
>
> On Wed, Nov 28, 2012 at 7:53 PM, Esteban Gonzalez
> <[log in to unmask] <mailto:[log in to unmask]>> wrote:
>
> Hi,
>
> I need to demonstrate that U(x) = phi(x) + x*Phi(x) > 0 for all x.
> This is clearly true for x >= 0. In addition U'(x) > 0 for all x,
> and thus it is a
> strictly increasing function.
>
> It would be sufficient to show, for instance that lim (x-> -
> infinity) U(x) = 0
>
> Any suggestion ?
>
> Thanks in advance,
> Esteban
>
>
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