What is the formula for the variance of a probability

David Troyer 2022-01-17 Answered
What is the formula for the variance of a probability distribution?
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Expert Answer

Lakisha Archer
Answered 2022-01-17 Author has 39 answers

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:
i=1nP(xi)=P(X=x)=P(x)=1
then the Expectation is defined by:
E(X)=i=1nxiP(X=xi)=xP(x)Then the variance is defined by:
Var(X)=E(X2){E(x)}2=i=1nxi2P(X=xi){i=1nxiP(X=xi)}2=x2P(x){xP(x)}2

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

P(aXb)=abf(x)dx=1

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

E(X)=abxf(x)dx=Dxf(x)dx

and the variance is defined by:
Var(X)=E(X2){E(X)}2=abx2f(x)dx{abxf(x)dx}2=Dx2f(x)dx{Dxf(x)dx}2

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