Variance and Covariance

  • Thread starter CescGoal
  • Start date
  • #1
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Very sorry that I've double posted but I realised i placed the original post in Precalculus.



1. Homework Statement
Question

Let X and Y be independent random variables with variances 9 and 7 respectively and let
Z = X - Y

a) What is the value of Cov(X,Z)
b) What is the value of the correlation coefficient of X and Z?

I've been stuck on this one question for 2-3 hours; its ridiculous, I know. Here's my terrible try.


3. The Attempt at a Solution
a)

Var(X) = 9
Var(Y) = 7

Var(X-Y) = Var(X) + Var(Y) = Var(Z)
Therefore, Var(Z) = 7 + 9 =16
Cov(X,Z) = E[XZ] - E[X]E[Z]


and b) [tex]\rho[/tex]XZ = [tex]\frac{Cov(X,Z)}{\sqrt{Var(X)*Var(Z)}}[/tex]



= [tex]\frac{Cov(X,Z)}{\sqrt{9}*\sqrt{16}}[/tex]
= [tex]\frac{Cov(X,Z)}{12}[/tex]

Since last topic, i've realised that Cov(X,Y) = 0 due to independency. But I don't know how to use it.
 

Answers and Replies

  • #2
Dick
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Cov(X,Z)=Cov(X,X-Y). Cov(X,X-Y)=Cov(X,X)-Cov(X,Y), right?
 
  • #3
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I didn't know about that law; thankyou very much.
 
  • #4
Dick
Science Advisor
Homework Helper
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I didn't know about that law; thankyou very much.
It's pretty obvious if you write out the definition of Cov. You should try and do that so you can see why it's true.
 

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