Combining $N$ number of normal distributions

Question

Suppose I have N normal distributions $N(u_{i},s_{i})$ for $i = 1\cdots N$. I want to combine all these distributions into one so that it serves as the average of all of them. How to do this? Is $u = \frac{u_{1}+u_{2}+...+u_{N}}{N}$ and $s = \sqrt\frac{s_{1}^2+s_{2}^2+...s_{N}^2}{N})$ correct?

Answer 1

Assuimng the $N(u_{i},s_{i})$ are independant for each $i$ we can compute the mean $u$ thanks to the more general result :

If $X_{1}, X_{2}, \cdots , X_{N}$ are mutually independent normal random variables with means $m_{1},\cdots,m_{N}$ and variance $\sigma_{1}^{2},\cdots, \sigma_{N}^{2}$ then the linear combination $Y = \sum\limits_{i=1}^{N}a_{i}X_{i}$ follows $N(\sum\limits_{i=1}^{N}a_{i}\mu_{i},\sum\limits_{i=1}^{N}a_{i}^{2}\sigma_{i}^{2})$