# What is the normalized product of two correlated Gaussian PDFs?

I want to simulate a sensor, which exhibits a Gaussian error distribution and temporal correlation $$\rho$$ between measurements. I want to have a formal description what fusing measurements with known correlation will do with the mean and variance, to say it bluntly.

The normalized product of two uncorrelated Gaussian PDFs can be written as:

$$\mu=\frac{\mu_1\sigma_2^2+\mu_2\sigma_1^2}{\sigma_1^2+\sigma_2^2},\qquad \frac1{\sigma^2}=\frac1{\sigma_1^2}+\frac1{\sigma_2^2}.$$ Source: Calculate the product of two Gaussian PDF's

According to: Generating time-correlated random noise with same start point I can use an AR(1) process to model the correlation between the Gaussians. I set $$n = 1$$ for one timestep. $$\epsilon_t$$ is uncorrelated Gaussian noise.

The AR(1) process: $$X_t = \varphi X_{t-1} + \epsilon_t$$

Variance: $$var(X_t) = \sigma^2 = \frac{\sigma_\epsilon^2}{1-\varphi^2}$$

Covariance: $$E[X_t X_{t+1}] = \frac{\sigma_\epsilon^2}{1-\varphi^2} \varphi = \sigma^2 \varphi$$

Correlation: $$\rho = \frac{E[X_t X_{t+1}]}{\sigma^2} = \frac{\sigma^2\varphi}{\sigma^2} = \varphi$$

1) Now, how do I fuse the correlated measurements $$X_t$$ and $$X_{t+1}$$?

If I do a Monte Carlo simulation in Octave/Matlab of say two fusions the RMSE of the error does not resemble the variance. But this is also dependent on how I set $$X_0$$ at $$t=1$$. Either I say:

-"This correlated process starts now, so I set $$X_{0} = 0$$ and start with $$var(X_1) = \epsilon_t$$."

or I say:

-"Even though I did not make any observations of this process, the first measurement will still be influenced by the hidden correlated stuff before. So, I can start off the bat with the regular formula: $$var(X_1) = \sigma^2$$." Here I would just have to simulate the process for some samples before.

2) What do you think is the correct way to simulate this process?