# What is the cross-correlation function of two gaussians

#### stuomiva

Hi everyone,

Earlier I posted a question regarding the probability of one gaussian distribution being similar to another one.

I might have stumbled on something that might answer the question but I'm not sure. I found out that the expectation of the difference of two gaussians can be calculated from their cross-correlation integral.

From Wolfram's Mathworld pages I found the convolution integral of two gaussians (which is the expectation of sum of two gaussians) but they didn't give explicit form of the cross-correlation of two gaussian functions.

Would this be the right way to calculate how similar two gaussians are?
If yes, does anyone know how to derive the cross-correlation of gaussians, or have the final answer at hand.

Cheers,

#### CaptainBlack

MHF Hall of Fame
Hi everyone,

Earlier I posted a question regarding the probability of one gaussian distribution being similar to another one.

I might have stumbled on something that might answer the question but I'm not sure. I found out that the expectation of the difference of two gaussians can be calculated from their cross-correlation integral.

From Wolfram's Mathworld pages I found the convolution integral of two gaussians (which is the expectation of sum of two gaussians) but they didn't give explicit form of the cross-correlation of two gaussian functions.

Would this be the right way to calculate how similar two gaussians are?
If yes, does anyone know how to derive the cross-correlation of gaussians, or have the final answer at hand.

Cheers,
The convolution integral of two Gaussians gives the distribution of a RV which is the sum of two RV each distributed as one of the Gaussians in the convolution.

Do you want to know the distribution of the difference between to Gaussian RVs?

If so you don't need convolutions or cross-correlations just knowlege:

Let $$\displaystyle X_1\sim N(\mu_1,\sigma_1^2)$$ and $$\displaystyle X_2\sim N(\mu_2,\sigma_2^2)$$

Then:

$$\displaystyle X_1-X_2 \sim N(\mu_1-\mu_1, \sigma_1^2+\sigma_2^2)$$

and for what its worth:

$$\displaystyle X_1+X_2 \sim N(\mu_1+\mu_1, \sigma_1^2+\sigma_2^2)$$

(However the above is not what is implied in the wording of your question/s)

CB