Any help with the following problem is appreciated (I have no idea how to approach it): Consider the integral . Express as a double integral involving and .
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Originally Posted by demode Any help with the following problem is appreciated: Consider the integral . Express as a double integral involving and . It's well known that . So if , surely ...
Originally Posted by Prove It It's well known that . So if , surely ... But the problem says "express as a double integral". How do we need to express it like that?
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since that is just the same integral with "dummy variable" y rather than x. Then by Fubini's theorem, that is the same as
Originally Posted by HallsofIvy since that is just the same integral with "dummy variable" y rather than x. Then by Fubini's theorem, that is the same as It should also be pointed out that this double integral can be evaluated to the value of by converting to polars...
Originally Posted by Prove It It should also be pointed out that this double integral can be evaluated to the value of by converting to polars... How did you evaluate it to the value of ? Here's my attempt: But how does this equal to ?
Last edited by demode; May 1st 2010 at 06:26 PM.
You know that . You can also say Therefore which means . Another handy application is using the Gaussian Integral to compute . Since is an even function, that means . Using the substitution yields and So that the integral becomes . So that means .
Originally Posted by demode How did you evaluate it to the value of ? Here's my attempt: But how does this equal to ? It was a double integral, remember? Let so that and . The integral becomes .
Originally Posted by HallsofIvy It was a double integral, remember? Let so that and . The integral becomes . Actually, I believe you'll find that . So the integral becomes
Prove It, We know that If I want to integrate the density function of the normal distribution: using the value of I, what would be a suitable change of variable in this case?
Originally Posted by demode Prove It, We know that If I want to integrate the density function of the normal distribution: using the value of I, what would be a suitable change of variable in this case? You can use the exact same change of variable as given to integrate the Gaussian function. . Since that means . We have shown that So that means . Therefore which is what we require for any probability density function.
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