Variance of a Random Variable

4. Probability Distributions: Random Variables

Variance of a Random Variable

Variance

Definition

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

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The standard deviation of $X$ is the positive square root of the variance. A benefit of the standard deviation is that it is expressed in the same units of measurements as $X$ .

Notation

$\sigma^2$

Alternatives: $\sigma_X^2$ or $Var[X]$

$\sigma$

Alternatives: $\sigma_X$ or $SD[X]$

Variance and Standard Deviation of a Discrete Random Variable

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

Then the variance of $X$ is calculated as follows:

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

$\sigma^2=\mathbb{E}[X^2] - (E[X])^2$
To calculate the standard deviation, simply take the positive square root of the variance:

$\sigma=\sqrt{\sigma^2}$

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

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

Calculate the variance and standard deviation of $X$ .

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

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

$\begin{array}{rcl} \mathbb{E}[X]&=&\sum\limits_{\text{all }x\text{ in }R(X)}x\cdot f(x)\\ &=&0 \cdot \mathbb{P}(X=0) + 1 \cdot \mathbb{P}(X=1) +2 \cdot \mathbb{P}(X=2) +3 \cdot \mathbb{P}(X=3) \\ &=&0 \cdot 0.3 + 1 \cdot 0.2 + 2 \cdot 0.1 + 3 \cdot 0.4\\ &=& 1.6\\ \end{array}$
Calculate $\mathbb{E}[X^2]$ :

$\begin{array}{rcl} \mathbb{E}[X^2]&=&\sum\limits_{\text{all }x\text{ in }R(X)}x^2\cdot f(x)\\ &=&0^2 \cdot \mathbb{P}(X=0) + 1^2 \cdot \mathbb{P}(X=1) +2^2 \cdot \mathbb{P}(X=2) +3^2 \cdot \mathbb{P}(X=3) \\ &=&0^2 \cdot 0.3 + 1^2 \cdot 0.2 + 2^2 \cdot 0.1 + 3^2 \cdot 0.4\\ &=& 4.2\\ \end{array}$
Calculate $\sigma^2$ :

$\begin{array}{rcl} \sigma^2 &=& \mathbb{E}[X^2] - (E[X])^2\\ &=& 4.2 - 1.6^2\\ &=& 1.64 \end{array}$
To calculate the standard deviation, take the positive square root of the variance:

$\sigma = \sqrt{\sigma^2} = \sqrt{1.64} = 1.281$

New example

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Variance and Standard Deviation of a Continuous Random Variable

The variance of a continuous random variable is computed using integral calculus and is beyond the scope of this course.