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

Suppose that X and Y are independent continuous random variables. Show that
${\sigma }_{x}y=0$
$\to$
${\sigma }_{xy}$
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Emma Cooper
The X and Y covariance can be defined as:
${\sigma }_{xy}=cov\left(X,Y\right)$
=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, ${\sigma }_{xy}=0$ for X and Y to be independent.