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Covariance - Properties

         

If the covariances between random variables \( X, \) \( Y, \) \( W,\) and \( Z \) are as follows: \[ \begin{matrix} \text{Cov}(X,Y) = 0.3& \text{Cov}(X,W) = 0.2 \\ \text{Cov}(X,Z) = 0 & \text{Cov}(Y,W) = 0.8 \\ \text{Cov}(Y,Z) = 0 & \text{Cov}(W,Z) = 0.4, \end{matrix} \] what is the covariance between \( 3X+2W \) and \( 8Y + 4Z? \)

If random variables \( X \) and \( Y \) have the following variance and covariance: \[ \text{Var}(X) = 6, \text{Var}(Y) = 8, \text{Cov}(X,Y) = 1, \] what is \( \text{Var}( 6X + 8Y ) \)?

If the covariance between random variables \( X \) and \( Y \) is \[ \text{Cov} (X,Y) = 0.3, \] what is \( \text{Cov} ( 4X , 2Y )? \)

If the covariance between random variables \( X \) and \( Y \) is \[ \text{Cov} (X,Y) = 0.2, \] what is \( \text{Cov} ( X + 6 , Y + 2 )? \)

Let \( X \) be a random variable uniformly distributed in the domain \( [-1,1], \) and let \( Y = X^2. \) What is the covariance between \( X \) and \( Y? \)

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