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Let the pdf defined as: $P(x, \bar{x}, \sigma) = \exp\left(\frac{-(x-\bar{x})^2}{2\sigma^2}\right)$. How can we integrate this probability density for some values of $x$ that are higher than a given value $y$ ? That is computing $P(x>y)$.

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    Do like suggested, at the end multiply answer obtained by $\sigma\sqrt{2\pi}$.2012-06-05

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What you claim to be a probability density function is in fact not a probability density function, and its integral will not give you a probability. $\begin{align*} \int_y^{\infty}\exp\left(\frac{-(x-\bar{x})^2}{2\sigma^2}\right)\;\mathrm dx &=\sigma\sqrt{2\pi}\int_y^{\infty}\frac{1}{\sigma\sqrt{2\pi}}\exp\left(\frac{-(‌​x-\bar{x})^2}{2\sigma^2}\right)\;\mathrm dx\\ &=\sigma\sqrt{2\pi}\left[1-\Phi\left(\frac{y-\bar{x}}{\sigma}\right)\right] \end{align*}$ where the value of $\Phi(\cdot)$ can be looked up in a table.

If you must write a computer program to compute (an approximation to) the value of the integral via explicit numerical integration, try some of the programs available in various computer languages at this site. Alternatively, if it is not a requirement that explicit numerical integration must be used, the approximate value of $\Phi(x)$ can be computed using the rational function approximations given in Section 26.2 of Abramowitz and Stegun. These approximations are used in most "scientific" calculators and even in MATLAB and similar packages. But if even greater accuracy is required, more accurate values can be obtained by summing the first few terms of one of the series for $\Phi(x)$ that are given in Chapter 26 of Abramowitz and Stegun. These methods are likely to be less time-consuming than explicit numerical integration.

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    The signification as you call it is that the probability that **any** normal random variable (with non-zero variance, to avoid trivial exceptions) exceeds _its_ mean $\mu_X$ by one of _its_ standard deviation $\sigma_X$ has value $1-\Phi(1)$. So you can make different choices of $\mu$ and $\sigma$ and torture a computer into doing repeated numerical integrations with different numbers, but you will get the _same_ value $1-\Phi(1) \approx 0.1587$ for P\{X > \mu_X + \sigma_X\}.2012-06-06