According to the summation property of Gaussian distributions: If there are 2 Gaussian distributions as given by:
$y$~$N (\mu, \Sigma)$ and $y'$~$N (\mu', \Sigma')$
then $y~+y'$~$N(\mu+ \mu', \Sigma+\Sigma')$
Consider 2 boxes that contain chits of papers with numbers written on them.
The first box contains chits numbered with either 1, 2 or 3 and written in a way, such that the probability of getting a number by randomly picking a chit from the box is a Gaussian distribution with $\mu_1=3$ and $\Sigma_1=1$.
Similarly, the numbers on the chits in box 2 can be either 4, 5 or 6, with a Gaussian distribution with $\mu_2=5$ and $\Sigma_2=1$
Now if I mix up the chits of both these boxes into one large box, then according to the summation rule, the new distribution has $\mu_3=7$ and $\Sigma=2$
Is this correct? It doesnt seem so because I dont have a chit with the nuber 7 on it, but the resultant distribution is telling me that getting a chit with 7 has the highest probability.