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

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Answer 1 · 2017-02-27T22:52:30

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:

$sum_(i=1)^n P(x_i) = sum P(X=x) = sum P(x) = 1$

then the Expectation is defined by:

$E(X) = sum_(i=1)^n \ x_i \ P(X=x_i) = sum xP(x)$

Then the variance is defined by:

$Var(X) = E(X^2) - {E(X)}^2$
$" " = sum_(i=1)^n \ x_i^2 \ P(X=x_i) - {sum_(i=1)^n \ x_i \ P(X=x_i)}^2$
$" " = sum x^2P(x) - {sum xP(x)}^2$

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

$P( a le X le b) = int_a^b \ f(x) \ dx = 1$

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

$E(X) = int_a^b \ xf(x) \ dx = int_D \ xf(x) \ dx$

and the variance is defined by:

$Var(X) = E(X^2) - {E(X)}^2$
$" " = int_a^b \ x^2f(x) \ dx - {int_a^b \ xf(x) \ dx}^2$
$" " = int_D \ x^2f(x) \ dx - {int_D \ xf(x) \ dx}^2$