# 17 Sums of Random Variables

## 17.1 Linear Transformation of a RV

Suppose that $$X$$ is a random variable and that $$a$$ and $$b$$ are constants. Then:

${\operatorname{E}}\left[a + bX \right] = a + b {\operatorname{E}}[X]$

${\operatorname{Var}}\left(a + bX \right) = b^2 {\operatorname{Var}}(X)$

## 17.2 Sums of Independent RVs

If $$X_1, X_2, \ldots, X_n$$ are independent random variables, then:

${\operatorname{E}}\left[ \sum_{i=1}^n X_i \right] = \sum_{i=1}^n {\operatorname{E}}[X_i]$

${\operatorname{Var}}\left( \sum_{i=1}^n X_i \right) = \sum_{i=1}^n {\operatorname{Var}}(X_i)$

## 17.3 Sums of Dependent RVs

If $$X_1, X_2, \ldots, X_n$$ are possibly dependent random variables, then:

${\operatorname{E}}\left[ \sum_{i=1}^n X_i \right] = \sum_{i=1}^n {\operatorname{E}}[X_i]$

${\operatorname{Var}}\left( \sum_{i=1}^n X_i \right) = \sum_{i=1}^n {\operatorname{Var}}(X_i) + \sum_{i \not= j} {\operatorname{Cov}}(X_i, X_j)$

Note that when $$X_i$$ and $$X_j$$ are independent ($$i \not= j$$), then $${\operatorname{Cov}}(X_i, X_j) = 0$$.

## 17.4 Means of Random Variables

Suppose $$X_1, X_2, \ldots, X_n$$ are independent and identically distributed (iid) random variables. Let $$\overline{X}_n = \frac{1}{n} \sum_{i=1}^n X_i$$ be their sample mean. Then:

${\operatorname{E}}\left[\overline{X}_n \right] = {\operatorname{E}}[X_i]$

${\operatorname{Var}}\left(\overline{X}_n \right) = \frac{1}{n}{\operatorname{Var}}(X_i)$