Julia White

2022-01-19

How can you find standard deviation from a probability distribution?

jean2098

Expert

Standard deviation $=\sqrt{E\left({X}^{2}\right)-{\left(E\left(X\right)\right)}^{2}}$
Explanation:
In a PDF, f(x), the expected mean is given by E(X)
Where $E\left(X\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x\cdot f\left(x\right)dx$
The variance is given by $Var\left(x\right)=E\left({X}^{2}\right)-{\left(E\left(X\right)\right)}^{2}$
Where $E\left(g\left(X\right)\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g\left(x\right)\cdot f\left(x\right)dx$
We know
Standard Deviation = $\sqrt{Variance}$
$⇒$ Standard deviation $=\sqrt{E\left({X}^{2}\right)-{\left(E\left(X\right)\right)}^{2}}$
Or...
$⇒$ Standard deviation $=\sqrt{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^{2}\cdot f\left(x\right)dx-{\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x\cdot f\left(x\right)dx\right)}^{2}}$

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