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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.

# 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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