Suppose that X and Y are independent continuous random variables. Show that sigma_(x)y=0 rightarrow sigma_(xy)

themediamafia73 2022-09-13 Answered
Suppose that X and Y are independent continuous random variables. Show that
σ x y = 0

σ x y
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Answers (1)

Emma Cooper
Answered 2022-09-14 Author has 9 answers
The X and Y covariance can be defined as:
σ x y = c o v ( X , Y )
=E[(X-E(X))][(Y-E(Y))]
=E[XY-YE(X)-XE(Y)+E(X)E(Y)]
=E(XY)-E(Y)E(X)-E(X)E(Y)+E(X)E(Y)
=E(XY)-E(X)E(Y)
Now, if X and Y are independent, then E(XY)=E(X)E(Y).
Therefore, σ x y = 0 for X and Y to be independent.

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