Question
Use expectations to show that Cov(X,Y) = E(XY) −E(X)E(Y).
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Answer to a math question Use expectations to show that Cov(X,Y) = E(XY) −E(X)E(Y).
Seamus
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To show that Cov(X,Y) = E(XY) - E(X)E(Y), we need to use the definitions of covariance and expectation.
The covariance between two random variables X and Y, denoted as Cov(X,Y), is given by:
Cov(X,Y) = E((X - \mu_X)(Y - \mu_Y))
where\mu_X \mu_Y
Using the linearity of expectation, we can expand the expression:
Cov(X,Y) = E(XY - \mu_XY - \mu_YX + \mu_X\mu_Y)
= E(XY) - E(\mu_XY) - E(\mu_YX) + E(\mu_X\mu_Y)
= E(XY) - \mu_XE(Y) - \mu_YE(X) + \mu_X\mu_Y
Since\mu_X \mu_Y
= E(XY) - \mu_XE(Y) - \mu_YE(X) + \mu_X\mu_Y
= E(XY) - \mu_XE(Y) - E(X)\mu_Y + E(X)\mu_Y
= E(XY) - \mu_XE(Y) - E(X)\mu_Y + E(X)\mu_Y
= E(XY) - \mu_XE(Y) - E(X)\mu_Y + E(X)\mu_Y
= E(XY) - E(X)\mu_Y - \mu_XE(Y) + E(X)\mu_Y
= E(XY) - E(X)\mu_Y - \mu_XE(Y) + E(X)\mu_Y
= E(XY) - E(X)E(Y) - E(X)E(Y) + E(X)E(Y)
= E(XY) - E(X)E(Y)
Therefore, Cov(X,Y) = E(XY) - E(X)E(Y).
\textbf{Answer:} Cov(X,Y) = E(XY) - E(X)E(Y)
The covariance between two random variables X and Y, denoted as Cov(X,Y), is given by:
where
Using the linearity of expectation, we can expand the expression:
Since
Therefore, Cov(X,Y) = E(XY) - E(X)E(Y).
\textbf{Answer:} Cov(X,Y) = E(XY) - E(X)E(Y)
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