Re: Gaussian distribution

On Mar 22, 5:34 pm, Kaba <REkalleMOunderscoreVErutane...@xxxxxxxxxxx>
wrote:
Hello

How do you prove that a Gaussian distribution integrates to 1 over -oo..
+oo?

That aside, I'm trying to find the fourier transform of a Gaussian
distribution, how do you do that?

My problems lie in the term of the form e^(bx^2) which I can't seem to
be able to integrate. I tried a change of variables x(u) = sqrt(u),
didn't take me anywhere. I also tried integration by parts two times
sequentially, didn't also take me anywhere.

This is not homework. I'd like to know the frequency response of a
Gaussian filter so I can design them using a desired cutoff frequency.

There's a trick to it.

Let I = integral(-inf, inf) exp(-x^2) dx

Multiply I by itself.

I^2 = [integral(-inf, inf) exp(-x^2) dx]*
[integral(-inf, inf) exp(-y^2) dy]

= integral(-inf,inf) integral(-inf,inf) exp[-(x^2+y^2)] dx dy

Now the double integral over dx dy can be changed to polar
coordinates. The exponential becomes exp(-r^2), but
because dx dy becomes r dr d(theta), a change of
variables to u=r^2 gives you an easy integral of
exp(-u) du

- Randy

.

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