Combining Multiple Normally Distributed Random Variables

When two independent random variables $X \sim N(\mu_X, \sigma_X^2)$ and $Y \sim N(\mu_Y, \sigma_Y^2)$ are added, the sum is normal with mean $\mu_X+\mu_Y$ and variance $\sigma_X^2+\sigma_Y^2$. The same idea extends to any finite number of independent normals.

Sum of independent normals. If $X_1, X_2, \ldots, X_n$ are independent and $X_i \sim N(\mu_i, \sigma_i^2)$, then

$$X_1 + X_2 + \cdots + X_n \;\sim\; N\!\left(\mu_1+\mu_2+\cdots+\mu_n,\ \sigma_1^2+\sigma_2^2+\cdots+\sigma_n^2\right).$$

Means add. Variances add. Standard deviations do not add.

For example, with $X_1 \sim N(2,1),\ X_2 \sim N(5,4),\ X_3 \sim N(-1,4)$ independent,

$$X_1+X_2+X_3 \;\sim\; N(2+5-1,\ 1+4+4) \;=\; N(6,9).$$

The standard deviation of the sum is $\sqrt{9}=3$, not $\sqrt{1}+\sqrt{4}+\sqrt{4}=5$.