I put this under statistics because of your knowledge of the Gaussian.(adsbygoogle = window.adsbygoogle || []).push({});

I have run into an elementary problem. I was considering what the average value, <x>, is for a Gaussian with an x offset, and got results which don't make sense to me.

First, it is obvious that for P1(x)=e^(-(x^2)) the average is 0. The integral of x*P1(x) from -inf to inf is zero.

So I decided to center the Gaussian at x=2: P2(x)=e^(-(x-2)^2). I was expecting an average of 2, but instead got 2 times the square root of pi! The integral of x*P2(x) from -inf to inf is 2*sqrt(pi)...

I thought, well perhaps this has to do with the infinities lying on either side, and the plus side starting at higher values than the minus side...So I decided to see the average for the Gaussian centered at 2 from x=0 to x=4. Surely that must be 2, it is clearly symmetric in that interval. Turns out that x*P2(x) from 0 to 4 is (according to Wolfram Alpha) still 2*sqrt(pi).

Can anyone explain why this last Gaussian's average x isn't 2?

Thanks.

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# Gaussian center not average value?

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