Question

Suppose that X is a random variable with mean 20 and standard deviation 5. Also suppose that Y is a random variable with mean 40 and standard deviation 10. Assume that the correlation between X and Y is 0.5. Find the mean and variance of the random variable Z = ????3X ???? 2Y . Be sure to show all your work.

Answer 1

Answer:

a

 $E(Z) = 202$  

b

$E(Z) = 198$

c

$E(Z) = -140$

Step-by-step explanation:

From the question we are told that

   The mean of X  is $E(X) = 20$

   The standard deviation of X is  $s_1 = 5$

   The mean of Y  is $\= y = 40$

    The standard deviation of Y is  $s_2 = 10$

Considering question a  

Generally the mean of   Z = 2 + 10X. is mathematically  represented

    $E(Z) = E[2 + 10X ]$

=> $E(Z) = 2 + 10 E(X)$

=>  $E(Z) = 2 + 10 * 20$        

=>  $E(Z) = 202$          

Considering question b

Generally the mean of   Z = 10X - 2.. is mathematically  represented  

        $E(Z) = E[10X -2 ]$

   => $E(Z) = 10E(X) - 2$

  =>  $E(Z) = 10* 20 - 2$

  =>  $E(Z) = 200 - 2$

 =>  $E(Z) = 198$

Considering question c

Generally the mean of   -3X - 2Y is mathematically  represented

     $E(Z) = E[-3X -2Y ]$

     $E(Z) = -3 E(X) -2E(Y)$

=>   $E(Z) = -3 * 20 -2* 40$

=> $E(Z) = -140$