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I am given two random variables $Y_1$ and $Y_2$ and:

  • $E(Y_1) = 4$
  • $E(Y_2) = -1$
  • $V(Y_1) = 2$
  • $V(Y_2) = 8$

I am asked to find $Cov(Y_1,Y_1)$

I know $Cov(Y_1,Y_2) = E(Y_1Y_2)-E(Y_1)E(Y_2)$

I'm not sure if it is a typo, but it says $Cov(Y_1,Y_1)$ NOT $Cov(Y_1,Y_2)$

The answer is $2$, but i'm not sure how to get $E(Y_1Y_2)$ from the information given, or why it's asking for the Cov of the same variable?

Thanks for any help!

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    You do not seem to have any information about the relationship between $Y_1$ and $Y_2$ and so cannot say anything about the covariance between them. It could be anywhere between $-4$ and $+4$2016-10-19

2 Answers 2

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$cov(Y_1,Y_1)=var(Y_1)=2$ This follows from the definition of covariance:

$cov(Y_1,Y_1)=E(Y_1\cdot Y_1)-E(Y_1)E(Y_1)=E(Y_1^2)-E(Y_1)^2$

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    love how it says that nowhere in my book... Thanks!2010-11-17
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    This follows from the definition. See above edit.2010-11-17
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Cov$(Y_1,Y_2) = \text{Cov}((Y_1-E[Y_1])(Y_2-E[Y_2])) = E[Y_1 Y_2]-E[Y_1]E[Y_2]$

Now let $Y_2 = Y_1$

Then Cov$(Y_1,Y_2) = E[{Y_1}^2] -(E[Y_1])^2$

This equation should look familiar...the variance of $Y_1$

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    nicely explained2010-11-17