Wierzycaz
2021-05-28
Answered

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

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Jaylen Fountain

Answered 2021-05-29
Author has **170** answers

Step 1

Consider the two normal random variables X and Y are independently and identically distributed.

$E(X)=E(Y)=\mu $

$V(X)=V(Y)={\sigma}^{2}$

Step 2

Consider,

$V(X)=E({X}^{2})-[E(X){]}^{2}$

$E({X}^{2})=V(X)-[E(X){]}^{2}$

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

$E({Y}^{2})={\sigma}^{2}-{\mu}^{2}\text{}\text{}\text{}[\because \text{}X\text{}and\text{}Y\text{}are\text{}identically\text{}distributed]$

Step 3

Now, the required quantity can be derived as follows:

$E(X-Y{)}^{2}=E({X}^{2}+{X}^{2}-2XY)$

$=E({X}^{2})+E({Y}^{2})-2E(X)E(Y)\text{}\text{}\text{}[\because \text{}X\text{}and\text{}Y\text{}are\text{}independent]$

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

$=2{\sigma}^{2}-4{\mu}^{2}$

Consider the two normal random variables X and Y are independently and identically distributed.

Step 2

Consider,

Step 3

Now, the required quantity can be derived as follows:

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