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Any solution for the following definite integaral? Here $\Phi(x)$ represents the cumulative distributive function of standard normal distribution $$\int_{\frac{-d}{\sqrt2\sigma}}^{\frac{d}{\sqrt2\sigma}} \frac{1}{2\sigma^2\sqrt{2\pi}} \left( \Phi\left(\alpha+\sqrt{d^2-\frac{x^2}{2\sigma^2}}\right)-\Phi\left(\alpha-\sqrt{d^2-\frac{x^2}{2\sigma^2}}\right)\right) \exp\left(\frac{-x^2}{2}\right) \; dx$$

The above function can be represented in terms of error function as $$\frac{1}{4\sigma^2\sqrt{2\pi}}\int_{\frac{-d}{\sqrt2\sigma}}^{\frac{d}{\sqrt2\sigma}}\exp\left(\frac{-x^2}{2}\right) \left(\text{erf}\left(\frac{\alpha+\sqrt{d^2-\frac{x^2}{2\sigma^2}}}{\sqrt{2}}\right)-\text{erf}\left(\frac{\alpha-\sqrt{d^2-\frac{x^2}{2\sigma^2}}}{\sqrt{2}}\right)\right)\; dx$$

any more help???

  • 0
    Do you mean bounds to be $\pm \sqrt{2} d \sigma$ instead of $\pm \frac{d}{\sqrt{2} \sigma}$ ?2011-09-01
  • 0
    No, I want bounds to be $\pm \frac{d}{\sqrt{2} \sigma}$2011-09-01

2 Answers 2