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How can I solve the following integral?

$$\int_{-\infty}^\infty \prod_{i=1}^n \bigg( 1 - \Phi\left(\frac{c - \mu_i}{\sigma_i}\right) \bigg) \frac{1}{\sigma_Y}\phi \bigg(\frac{c-\mu_Y}{\sigma_Y} \bigg) \,\mathrm dc$$

where Φ(⋅) is the CDF of Normal distribution and ϕ(⋅) is the pdf of Normal distribution.

even when $n=1$ I cannot solve it. I would appreciate if you could help me either for the case that $n=1$ or the above general case.

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    $\begingroup$ I know this is from statistics, but you may want to define your terms for the nonspecialists who still may be able to help you. $\endgroup$ Commented Feb 5, 2013 at 1:12
  • $\begingroup$ IF $\Phi(\cdot)$ is the CDF of Normal RV and $\phi(\cdot)$ is the pdf of Normal RV, did you try $\frac{1}{\sigma} \phi \bigg( \frac{c-\mu}{\sigma} \bigg)dc=d \Phi \bigg(\frac{c-\mu}{\sigma} \bigg)$ $\endgroup$ Commented Feb 5, 2013 at 1:34
  • $\begingroup$ thank you I edited the question $\endgroup$ Commented Feb 5, 2013 at 1:38
  • $\begingroup$ No I did not try this. How can this substitution solve this integral? $\endgroup$ Commented Feb 5, 2013 at 1:44
  • $\begingroup$ Got something from my answer below? $\endgroup$ Commented Feb 9, 2013 at 11:31

1 Answer 1

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Considering independent random variables $X_i$ and $Y$, gaussian with mean and variance $(\mu_i,\sigma_i^2)$ and $(\mu_Y,\sigma_Y^2)$ respectively, this integral is $$ \mathbb P(\min\limits_{1\leqslant i\leqslant n}X_i\geqslant Y). $$ If $n=1$, its value is $$ \Phi\left(\frac{\mu_1-\mu_Y}{\sqrt{\sigma_1^2+\sigma_Y^2}}\right). $$ For $n\geqslant2$, I see no reason to expect some simple explicit formulas in the general case.

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