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The bivariate random variables $X$ and $Y$ have joint probability distribution specified by the following table:

	$y$ =1	$y$ =2	$y$ =3	$y$ =4
$x$ =1	0.05	0.10	0.10	0.01
$x$ =2	0.15	0.25	0.20	0.05
$x$ =3	0.01	0.03	0.04	0.01

Please provide all answers to the following to three decimal places.

(a) Find the expectation of $XY$ .

(b) Find the covariance Cov( $X,Y$ ) between $X$ and $Y$ .

(c) What is the correlation between $X$ and $Y$ ?

(d) Suppose the random variables $X$ and $Y$ above are connected to random variables $U$ and $V$ by the relations

$X = 3 U + 6$
$Y = 3 V + 8$

What is the covariance Cov( $U,V$ )?

(e) What is the correlation between $U$ and $V$ ?

You can earn partial credit on this problem.