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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.

1 Answer

1 vote

Answer:

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

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