STATISTICS : Sum of two distributions

Question

Let $\Omega$ represent some physical property such that, $\Omega\sim\mathcal{N}(\mu,\sigma)$. Given that some computational method exists that does not deal with $\Omega$ directly but rather deals with the change in $\Omega$, that is, $\delta\Omega$. If after computation the updated distribution in $\delta\Omega$ is such that, $\delta\Omega\sim\mathcal{N}(\mu_0,\sigma_o)$. How would one go about computing the statistics of updated $\Omega^*$ such that: $$\Omega^* =\Omega + \delta\Omega$$

Answer 1

The sum of two independent normally distributed random variables $\sim \mathcal{N}(\mu,\sigma^2)$ and $\mathcal{N}(\mu_o,\sigma_o^2)$ is a normally distributed random variable $\sim \mathcal{N}(\mu+\mu_o,\sigma^2+\sigma_o^2)$

You can show this using the probability density functions or the characteristic funtions