i need to calculate some expectations which involving the ratio of normal pdf to normal cdf.

Specifically, they are $E\{\phi(x)/\Phi(x)\}$ and $E\{x\phi(x)/\Phi(x)\}$ where $x\sim N(0,1)$.

Written in integral, they should be $\int_{-\infty }^{+\infty }{\dfrac {\phi^2 \left( x\right) } {\Phi \left( x\right) }dx}$ and $\int_{-\infty }^{+\infty }{\dfrac {x\phi^2 \left( x\right) } {\Phi \left( x\right) }dx}$, respectively.

I know these can be numerically evaluated, for example, by the integral function in Matlab.

My question is that, can these expectations be evaluated analytically, or be approximated in closed-form?

  • $\begingroup$ Are those the standard normal PDF and CDF, i.e., mean 0 and variance 1? $\endgroup$ – horchler Aug 6 '13 at 15:59
  • $\begingroup$ Yes, they are standard pdf and cdf with mean 0 and variance 1. $\endgroup$ – Sheng2012 Aug 6 '13 at 16:10

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