Linda Seales

2022-01-11

If X and Y are independent and identically distributed with mean $\mu$ and variance ${\sigma }^{2}$, find $E\left[{\left(X-Y\right)}^{2}\right]$

Joseph Fair

Expert

X and Y are independent and identically distribuded with mean $\mu$ and variance ${\sigma }^{2}$,
$E\left[X\right]=E\left[Y\right]=\mu$
$Var\left(X\right)=Var\left(Y\right)={\sigma }^{2}$
$Var\left(X\right)=E\left[{X}^{2}\right]-{\left(E\left[X\right]\right)}^{2}$
$E\left[{X}^{2}\right]=Var\left(X\right)+{\left(E\left[X\right]\right)}^{2}$
$={\sigma }^{2}+{\mu }^{2}$
X and Y are independent,
$E\left[{\left(X-Y\right)}^{2}\right]=E\left[{X}^{2}-2XY+{Y}^{2}\right]$
$=E\left[{X}^{2}\right\}-2E\left[X\right]E\left[Y\right]+E\left[{Y}^{2}\right]$
$=\left({\sigma }^{2}+{\mu }^{2}\right)-2\mu \mu +\left({\sigma }^{2}+{\mu }^{2}\right)$
$=2{\sigma }^{2}+2{\mu }^{2}-2{\mu }^{2}$
$=2{\sigma }^{2}$
Finally, $E\left[{\left(X-Y\right)}^{2}\right]=2{\sigma }^{2}$

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