# What is the formula for the variance of a probability

What is the formula for the variance of a probability distribution?
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Lakisha Archer

Often a shorthand notation is used, where the limits of summation (or integration) are omitted, and sometimes the subscripts;
For a discrete Random Variable X, where:
$\sum _{i=1}^{n}P\left({x}_{i}\right)=\sum P\left(X=x\right)=\sum P\left(x\right)=1$
then the Expectation is defined by:
$E\left(X\right)=\sum _{i=1}^{n}{x}_{i}P\left(X={x}_{i}\right)=\sum xP\left(x\right)$Then the variance is defined by:
$Var\left(X\right)=E\left({X}^{2}\right)-{\left\{E\left(x\right)\right\}}^{2}\phantom{\rule{0ex}{0ex}}=\sum _{i=1}^{n}{x}_{i}^{2}P\left(X={x}_{i}\right)-{\left\{\sum _{i=1}^{n}{x}_{i}P\left(X={x}_{i}\right)\right\}}^{2}\phantom{\rule{0ex}{0ex}}=\sum {x}^{2}P\left(x\right)-{\left\{\sum xP\left(x\right)\right\}}^{2}$

We get similar results for a continuous Random Variable X, where:

$P\left(a\le X\le b\right)={\int }_{a}^{b}f\left(x\right)dx=1$

and the Expectation, is defined by (and shorthand):

$E\left(X\right)={\int }_{a}^{b}xf\left(x\right)dx={\int }_{D}xf\left(x\right)dx$

and the variance is defined by:
$Var\left(X\right)=E\left({X}^{2}\right)-{\left\{E\left(X\right)\right\}}^{2}\phantom{\rule{0ex}{0ex}}={\int }_{a}^{b}{x}^{2}f\left(x\right)dx-{\left\{{\int }_{a}^{b}xf\left(x\right)dx\right\}}^{2}\phantom{\rule{0ex}{0ex}}={\int }_{D}{x}^{2}f\left(x\right)dx-{\left\{{\int }_{D}xf\left(x\right)dx\right\}}^{2}$