I.I.D Normal Random Variables

A sequence of random variables $X_1, X_2, \ldots, X_n$ is independent and identically distributed (abbreviated i.i.d.) if:

1. They are mutually independent.
2. They all share the same distribution.

When the common distribution is normal with mean $\mu$ and variance $\sigma^2$, we write

$$X_1, X_2, \ldots, X_n \overset{\text{i.i.d.}}{\sim} N(\mu, \sigma^2).$$

Because sums of independent normals are normal, the sum of i.i.d. normals is normal as well. Specifically, if $X_1, \ldots, X_n \overset{\text{i.i.d.}}{\sim} N(\mu, \sigma^2)$, then

$$S_n = X_1 + X_2 + \cdots + X_n \sim N(n\mu,\, n\sigma^2).$$

The mean and the variance each get multiplied by $n$.

To illustrate, suppose $X_1, X_2, X_3 \overset{\text{i.i.d.}}{\sim} N(7, 5)$. Then

$$\begin{align*} E[S_3] &= 3 \cdot 7 = 21, \\ \text{Var}(S_3) &= 3 \cdot 5 = 15, \end{align*}$$

so $S_3 \sim N(21, 15)$.