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

a

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

b

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

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 = √(V(C))`

substituting values

`\sigma = √(96)`

`\sigma = 9.798`