IB Maths AA 4.8 Notes

Variance of a discrete random variable

This section extends earlier work on random variables by introducing variance for discrete random variables, continuous random variables and their probability density functions, and the effect of linear transformations.

AS a reminder, a discrete random variable $X$ with possible values $x$ and probabilities $P(X=x)$, the mean or expected value is

$E(X)=\sum xP(X=x)$

To measure spread, we use the variance.

$\mathrm{Var}(X)=E(X^2)-[E(X)]^2$

where $E(X^2)=\sum x^2P(X=x)$.

The standard deviation is

$\sigma=\sqrt{\mathrm{Var}(X)}$

Find the standard deviation for the following distribution.

$x$	$0$	$1$	$2$
$P(X=x)$	$0.2$	$0.5$	$0.3$

First find the mean: $E(X)=0(0.2)+1(0.5)+2(0.3)=0+0.5+0.6=1.1$.

Now find $E(X^2)=0^2(0.2)+1^2(0.5)+2^2(0.3)=0+0.5+1.2=1.7$.

So $\mathrm{Var}(X)=1.7-(1.1)^2=1.7-1.21=0.49$.

Hence the standard deviation is $\sqrt{0.49}=0.7$.