# Gaussian difference distribution

let $f_1(x,y)$ and $f_2(x,y)$ be two 2-dimensional Gaussian functions with means $(\mu_{1_x},\mu_{1_y}$} & $(\mu_{2_x},\mu_{2_y}$} and variances $(\sigma_{1_x},\sigma_{1_y}$} & $(\sigma_{2_x},\sigma_{2_y}$} respectively. Assuming that the Gaussian functions are not infinite , for example

$$f(x,y)=\frac{1}{2\pi\sigma^2}\int_{-a}^{a}\int_{-b}^{b}\exp\big(-({(x-\mu_x)^2+(y-\mu_y)^2})/{2\sigma^2}\big) \,\mathrm{d}x\,\mathrm{d}y$$

I am interested in finding the probability that the function $f_1$ lies within some distance $d$ of $f_2$, i.e, Pr(dist($f_1$,$f_2$)$\leq d$). Is there any way to find a Gaussian difference distribution expression for some finite Gaussian functions?

http://mathworld.wolfram.com/NormalDifferenceDistribution.html gives the Gaussian difference distribution of infinite Gaussian, but I am interested in bounded Gaussian. Please help.

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The value of your displayed double integral is a number, and so $f(x,y)$ has the same value for all $x$ and $y$. How then is it a two-dimensional Gaussian function? –  Dilip Sarwate May 18 '12 at 11:38
Assuming that there is no correlation, 2D Gaussian is simply the multiplication of 2 1D Gaussian functions. Here I am using the same assumption. –  shaikh May 18 '12 at 11:51
If $\mu_x=\mu_y=0$, $\sigma^2=1$, and $a= b=1$, your $f(x,y)$ as given by the displayed integral has value $(\Phi(1)-\Phi(-1))^2 \approx 0.4659$. So what is the meaning of $f(x,y) = 0.4659$ and how, if at all, is $f(x,y)$ related to $f_1(x,y)$ and $f_2(x,y)$? –  Dilip Sarwate May 18 '12 at 12:27
Isn't if central probability of 2D Gaussian? If I am wrong please help me. I am interested in finding the probability that the function $f_1$ lies within some distance d of $f_2$, where $f_1$ and $f_2$ are two 2D Gaussian functions. If the above expression of $f(x,y)$ is wrong then please suggest me correct one. Please also refer math.stackexchange.com/questions/143377/… –  shaikh May 21 '12 at 1:04