If X and Y are independent and identically distributed with mean μ and variance σ2,...

Linda Seales

Linda Seales

Answered

2022-01-11

If X and Y are independent and identically distributed with mean μ and variance σ2, find E[(XY)2]

Answer & Explanation

Joseph Fair

Joseph Fair

Expert

2022-01-12Added 34 answers

X and Y are independent and identically distribuded with mean μ and variance σ2,
E[X]=E[Y]=μ
Var(X)=Var(Y)=σ2
Var(X)=E[X2](E[X])2
E[X2]=Var(X)+(E[X])2
=σ2+μ2
X and Y are independent,
E[(XY)2]=E[X22XY+Y2]
=E[X2}2E[X]E[Y]+E[Y2]
=(σ2+μ2)2μμ+(σ2+μ2)
=2σ2+2μ22μ2
=2σ2
Finally, E[(XY)2]=2σ2

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