Sum of Two Independent Normals

Means of independent random variables add directly.

Variances of independent random variables also add; the sum of independent normals is normal.

1. 1. Add the means

   For independent normals, $E[X+Y]=E[X]+E[Y]=2+3=5$.

2. 2. Add the variances

   Independence gives $\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)=4+9=13$, and a sum of independent normals is normal.

   $$X+Y\sim N(5,13)$$

Answer: N(5,13)