Covariance question

**gklove56** · April 30th 2010, 02:45 PM

please please help me out for these question below. They are very important to me. As I am not good on Statistic and I don't have the model answers for them, I really want someone who can show me the answers about them. Therefore, I can check if i was doing the questions in the right way.

Thank you very much!!!

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Question 1 -- Let X, Y and Z be random variables.
(a) Define the covariance Cov(X, Y ).
(b) Prove that Cov(Z,X + Y ) = Cov(Z,X) + Cov(Z, Y ).
(c) Suppose that X and Y are independent, with Var(X) = 25 and Var(Y ) = 6,
and that Z = X + 2Y . Find the correlation between X and Z.

**harish21** · April 30th 2010, 10:55 PM

Originally Posted by gklove56

please please help me out for these question below. They are very important to me. As I am not good on Statistic and I don't have the model answers for them, I really want someone who can show me the answers about them. Therefore, I can check if i was doing the questions in the right way.

Thank you very much!!!

================================================== =======
Question 1 -- Let X, Y and Z be random variables.
(a) Define the covariance Cov(X, Y ).
(b) Prove that Cov(Z,X + Y ) = Cov(Z,X) + Cov(Z, Y ).
(c) Suppose that X and Y are independent, with Var(X) = 25 and Var(Y ) = 6,
and that Z = X + 2Y . Find the correlation between X and Z.

(a) $Cov(X,Y) = E[({X_1}-{\mu_x})({Y}-{\mu_y})]$

$= E[XY] - {\mu_y}E(X) - {\mu_x}E(Y) + {\mu_x}{\mu_y}$

$=E[XY] - {\mu_y}{\mu_x}- {\mu_x}{\mu_y}+{\mu_y}{\mu_y}$

$\therefore Cov(X,Y)= E[XY] - {\mu_x}{\mu_y} = E[XY] - E[X]E[Y]$

After knowing (a), you can solve (b) as:

$Cov(Z,X+Y) = E[Z(X+Y)] - E[Z]E[X+Y]$

$= E[ZX]+E[ZY] - E[Z] {E[X]+E[Y]}$

$= E[ZX]+E[ZY] - E[Z]E[X] - E[Z]E[Y]$

$= \{E[ZX]-E[Z]E[X]\} + \{E[ZY] - E[Z]E[Y]\}$

$= Cov(Z,X) + Cov(Z,Y)$

(c) refer to the definition of correlation( $\rho$ ) in your book/notes

**gklove56** · May 1st 2010, 04:57 AM

Thank you so much, you are great!!!

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