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

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Lakisha Archer · Accepted Answer

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:
$E (X) = \sum_{i = 1}^{n} x_{i} P (X = x_{i}) = \sum x P (x)$ Then the variance is defined by:
$V a r (X) = E (X^{2}) - {E (x)}^{2} = \sum_{i = 1}^{n} x_{i}^{2} P (X = x_{i}) - {\sum_{i = 1}^{n} x_{i} P (X = x_{i})}^{2} = \sum x^{2} P (x) - {\sum x P (x)}^{2}$

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

$P (a \leq X \leq b) = \int_{a}^{b} f (x) d x = 1$

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

$E (X) = \int_{a}^{b} x f (x) d x = \int_{D} x f (x) d x$

and the variance is defined by:
$V a r (X) = E (X^{2}) - {E (X)}^{2} = \int_{a}^{b} x^{2} f (x) d x - {\int_{a}^{b} x f (x) d x}^{2} = \int_{D} x^{2} f (x) d x - {\int_{D} x f (x) d x}^{2}$

What is the formula for the variance of a probability

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