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Variance

Would you rather get paid $2 for flipping heads, or $6 for rolling a "1"? The expected value is the same ($1)...but the bets are different! Variance and standard deviation add color to probability.

Properties of Covariance

         

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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