Variance of a Random Variable

Chapter 4. Probability Distributions: Random Variables

Variance of a Random Variable

Variance and Standard Deviation

Definition

The variance of a random variable $X$ is the average squared deviation from its expected value.

The standard deviation of $X$ is the positive square root of the variance.

Notation

$\begin{array}{c}\mathbb{V}[X]\,\,\,\,\text{or}\,\,\,\,\sigma^2 \\\\ \mathbb{SD}[X]\,\,\,\,\text{or}\,\,\,\,\sigma \end{array}$

Let $X$ be a discrete random variable with expected value $\mathbb{E}[X]$ , where:

$\mathbb{E}[X] = \displaystyle\sum_{\text{all }x\text{ in }R(X)}x\cdot \mathbb{P}(X=x)$
Then the variance of $X$ is calculated as follows:

$\mathbb{V}[X] = \mathbb{E}[(X - \mathbb{E}[X])^2]$
Which can be rewritten in a way that is easier to compute:

$\begin{array}{rcl} \mathbb{V}[X] &=& \mathbb{E}[X^2] - (\mathbb{E}[X])^2 \\\\ &&\text{where} \\\\ \mathbb{E}[X^2] &=& \displaystyle\sum_{\text{all }x\text{ in }R(X)}x^2\cdot \mathbb{P}(X=x) \end{array}$

To calculate the standard deviation, simply take the positive square root of the variance:

$\mathbb{SD}[X]=\sqrt{\mathbb{V}[X]}$

Consider the following probability distribution of a discrete variable $X$ :

$x$	$0$	$1$	$2$	$3$
$\mathbb{P}(X=x)$	$0.2$	$0.3$	$0.1$	$0.4$

Calculate the variance and standard deviation of $X$ .

To calculate the variance of $X$ , we use the following formula:

$\mathbb{V}[X]=\mathbb{E}[X^2] - (\mathbb{E}[X])^2$
Calculate $\mathbb{E}[X]$ :

$\begin{array}{rcl} \mathbb{E}[X]&=&\sum\limits_{\text{all }x\text{ in }R(X)}x\cdot \mathbb{P}(X=x)\\\\ &=&0 \cdot \mathbb{P}(X=0) + 1 \cdot \mathbb{P}(X=1) +2 \cdot \mathbb{P}(X=2) +3 \cdot \mathbb{P}(X=3) \\\\ &=&0 \cdot 0.2 + 1 \cdot 0.3 + 2 \cdot 0.1 + 3 \cdot 0.4\\\\ &=& 1.7\\ \end{array}$
Calculate $\mathbb{E}[X^2]$ :

$\begin{array}{rcl} \mathbb{E}[X^2]&=&\sum\limits_{\text{all }x\text{ in }R(X)}x^2\cdot \mathbb{P}(X=x)\\\\ &=&0^2 \cdot \mathbb{P}(X=0) + 1^2 \cdot \mathbb{P}(X=1) +2^2 \cdot \mathbb{P}(X=2) +3^2 \cdot \mathbb{P}(X=3) \\\\ &=&0^2 \cdot 0.2 + 1^2 \cdot 0.3 + 2^2 \cdot 0.1 + 3^2 \cdot 0.4\\\\ &=& 4.3\\ \end{array}$
Calculate $\mathbb{V}[X]$ :

$\begin{array}{rcl} \mathbb{V}[X] &=& \mathbb{E}[X^2] - (\mathbb{E}[X])^2\\\\ &=& 4.3 - 1.7^2\\\\ &=& 1.41 \end{array}$
To calculate the standard deviation, take the positive square root of the variance:

$\begin{array}{rcl} \mathbb{SD}[X] &=& \sqrt{\mathbb{V}[X]} \\\\ &=& \sqrt{1.41} \\\\ &=& 1.187 \end{array}$

New example

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Variance and Standard Deviation of a Continuous Random Variable
The computation of the variance of a continuous random variable is done using integral calculus and is beyond the scope of this course.