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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

         

If the joint probability distribution function of \( X \) and \( Y \) is given as follows, what is \( 20 \text{Cov}(X,Y) ? \)

\[ \begin{matrix} p(X,Y) & X=0 & X=1 & X=2 \\ Y=0 & 0.1 & 0.1 & 0.2 \\ Y=1 & 0.2 & 0.2 & 0 \\ Y=2 & 0 & 0.1 & 0.1 \end{matrix} \]

For random variables \(X\) and \(Y,\) which of the following is the same as \[ \text{Cov}(X,Y)? \]

If the joint probability density function of continuous random variables \( X \) and \( Y \) is \[ f(x,y) = \frac{1}{6} \left( 8 x + 4 y \right) \] with \( 0 \le x \le 1 \text{ and } 0 \le y \le 1, \) what is the covariance between \( X \) and \( Y? \)

If \(X\) and \(Y\) are independent random variables where \(E[X]=10 \) and \( E[Y]=8, \) what is \(E[XY]?\)

If the probability distributions of continuous random variables \( X \) and \( Y \) are independent of each other, which of the following is equal to \( \text{Cov}(X,Y)? \)

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