\(X\sim N(\mu_{1},\sigma_{1}^{2})\)

\(Y\sim N(\mu_{1}, \sigma_{2}^{2})\)

X and Y are independent.

\(N(\mu_{1},\sigma_{1}^{2})+N(\mu_{1},\sigma_{2}^{2})\sim N(\mu_{1}+\mu_{2},\sigma_{1}^{2}+\sigma_{2}^{2})\)

So,

\(X+Y\) follows normal distribution

\((X+Y)\sim N(\mu_{1}+\mu_{2}, \sigma_{1}^{2}+\sigma_{2}^{2})\)