of the Gaussian/Normal
APPLICATIONS
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Gaussians can be used to integrate exponentials of the form: Below are some examples showing how it can be done.
Case I:
Example:
Therefore If this were a normal distribution, the constant in front would not be 1, it would be , so we need to make that constant appear in front:
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Case II: General:
Example:
- The constant a still determines the value of sigma as in Case 1.
therefore
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- Now we need to be concerned with the x and constant terms as they are related to the still undetermined m. The quadratic must be converted to the form (x+m)2 using the method of completing the square.
therefore and m=-1/4.
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- So the original integral becomes
which after adjusting the constants becomes
- For a discussion on the erf() function, see the page on the cumulative distribution function (CDF) of an Gaussian random variable
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