Sums of Random Variables

4. Probability Distributions: Random Variables

Sums of Random Variables

Sum of Independent Random Variables

Suppose two independent random variables $Y$ and $Z$ are added to give a new variable $X$ . Then their sum is also a random variable, with the following expected value and variance:

$\mathbb{E}[X] = \mathbb{E}[Y] + \mathbb{E}[Z]$
$Var[X] = Var[Y] + Var[Z]$

If, in contrast, $Y$ and $Z$ are subtracted, then the expected value and variance new variable $X$ are

$\mathbb{E}[X] = \mathbb{E}[Y] - \mathbb{E}[Z]$
$Var[X] = Var[Y] + Var[Z]$

If a random variable $Y$ is multiplied with a constant $p$ , the resulting variable $X$ has the following properties.

$\mathbb{E}[X] = p \cdot \mathbb{E}[Y]$
$Var[X] = p^2 \cdot Var[Y]$
$SD[X] = |p| \cdot SD[Y]$

Suppose:

$\,\,\,\,\,\,\,\,\scriptsize{\bullet}$ $\,\mathbb{E}[Y]= 6.2\phantom{0}$ $\,\,\,\,\,\,\,\,\scriptsize{\bullet}$ $\,Var[Y]= 2.37$

$\,\,\,\,\,\,\,\,\scriptsize{\bullet}$ $\,\mathbb{E}[Z]= 6.1\phantom{0}$ $\,\,\,\,\,\,\,\,\scriptsize{\bullet}$ $\,Var[Z]= 2.50$

$\,\,\,\,\,\,\,\,\scriptsize{\bullet}$ $\,X= 5Y -Z$

Find $\mathbb{E}[X]$ .

$\mathbb{E}[X]=24.9$

To calculate $\mathbb{E}[X]$ , make use of the following properties:

If $X = Y+Z$ , then $\mathbb{E}[X] = \mathbb{E}[Y] + \mathbb{E}[Z]$ .
If $X = k\cdot Y$ , then $\mathbb{E}[X] = k \cdot \mathbb{E}[Y]$ .

Applying these properties to $X= 5Y -Z$ , we get:

$\begin{array}{rcl} \mathbb{E}[X] &=& \mathbb{E}[5Y -Z]\\ &=& \mathbb{E}[5Y] + \mathbb{E}[-Z]\\ &=& 5\cdot \mathbb{E}[Y] -1 \cdot \mathbb{E}[Z]\\ &=& 5\cdot 6.2 -1 \cdot6.1\\ &=& 24.9 \end{array}$

New example

Sum of Dependent Random Variables

Suppose two dependent random variables $Y$ and $Z$ are added to give a new variable $X$ . Then their sum is also a random variable, with the following expected value and variance:

$\mathbb{E}[X] = \mathbb{E}[Y] + \mathbb{E}[Z]$
$Var[X] = Var[Y] + Var[Z] + 2Cov[Y,Z]$

If, in contrast, $Y$ and $Z$ are subtracted, then the expected value and variance new variable $X$ are

$\mathbb{E}[X] = \mathbb{E}[Y] - \mathbb{E}[Z]$
$Var[X] = Var[Y] + Var[Z] - 2Cov[Y,Z]$

The above behaviour follows form the general rule for adding variablels $Y$ and $Z$ multiplied by respectively constants $p$ and $q$ to give a new variable $X$ :

$\mathbb{E}[X] = p \cdot \mathbb{E}[Y] + q \cdot \mathbb{E}[Z]$
$Var[X] = p^2 \cdot Var[Y] + q^2 \cdot Var[Z] + 2pq \cdot Cov[Y,Z]$