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

Wierzycaz 2021-05-28 Answered
If X and Y are independent and identically distributed with mean μ and variance σ2, find E[(XY)2]
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Expert Answer

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)=μ
V(X)=V(Y)=σ2
Step 2
Consider,
V(X)=E(X2)[E(X)]2
E(X2)=V(X)[E(X)]2
E(X2)=σ2μ2
E(Y2)=σ2μ2   [ X and Y are identically distributed]
Step 3
Now, the required quantity can be derived as follows:
E(XY)2=E(X2+X22XY)
=E(X2)+E(Y2)2E(X)E(Y)   [ X and Y are independent]
=σ2μ2+σ2μ22μ2
=2σ24μ2
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