Linda Seales

2022-01-11

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

Joseph Fair

Beginner2022-01-12Added 34 answers

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,

$E\left[{(X-Y)}^{2}\right]=E[{X}^{2}-2XY+{Y}^{2}]$

$=E\left[{X}^{2}\right\}-2E\left[X\right]E\left[Y\right]+E\left[{Y}^{2}\right]$

$=({\sigma}^{2}+{\mu}^{2})-2\mu \mu +({\sigma}^{2}+{\mu}^{2})$

$=2{\sigma}^{2}+2{\mu}^{2}-2{\mu}^{2}$

$=2{\sigma}^{2}$

Finally,$E\left[{(X-Y)}^{2}\right]=2{\sigma}^{2}$

X and Y are independent,

Finally,

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