Question

An electronic system has each of two different types of components in joint operation. Let X and Y denote the lengths of life, in hundreds of hours, for components of type I and type II, respectively. The performance of a component is independent for each other. Let E(X) = 4, E(Y) = 2, E(X2) = 24, E(Y2) = 8.
The cost of replacing the two components depends upon their length of life at failure and it is given by C = 50 + 2X + 4Y.
(i) Compute the average cost of replacing the two components. Your final answer must be a number.
(ii) Compute the standard deviation of cost of replacing the two components. Your final answer must be a number.

Answer 1

Answer:

a

The average cost is $E(C) = 66$

b

The standard deviation of  cost is  $\sigma = 9.798$

Step-by-step explanation:

From the question we are told that

          $E(X) = 4$

          $E(Y) = 2$

          $E(X^2) = 24$

           $E(Y^2) = 8$

The cost of replacing the two component is  C =  50 + 2 X + 4 Y

The variance   of X is mathematically represented as

           V(X) =  $E(X^2) - [E(X)]^2$

Substituting values  

              $V[X] = 24 - 4^2$

               $V[X] =8$

The variance  of Y is mathematically represented as

           V(Y) =  $E(Y^2) - [E(Y)]^2$

Substituting values  

              $V[Y] = 8 - 2^2$

               $V[X] =4$

The average of  replacing the two component is  

        $E(C) = 2 * E(X) + 4* E(Y)$

substituting value

         $E(C) = 50 + 2 * (4) + 4* (2)$

         $E(C) = 66$

The variance  of replacing the two component is  

        $V(C) = V(50 + 2X +4Y)$      Note: The variance of constant is zero

                                                                  and X and  Y are independent  

=>      $V(C) = 2^2 * V(X) + 4^2 * V(Y)$

substituting values

=>       $V(C) = 4 * 8 + 16 * 4$        

=>         $V(C) = 32 + 64$

=>         $V(C) = 96$

The standard deviation is

         $\sigma = \sqrt{V(C)}$  

substituting values

           $\sigma = \sqrt{96}$

           $\sigma = 9.798$