Answer:
a
The average cost is [tex]E(C) = 66[/tex]
b
The standard deviation of cost is [tex]\sigma = 9.798[/tex]
Step-by-step explanation:
From the question we are told that
[tex]E(X) = 4[/tex]
[tex]E(Y) = 2[/tex]
[tex]E(X^2) = 24[/tex]
[tex]E(Y^2) = 8[/tex]
The cost of replacing the two component is C = 50 + 2 X + 4 Y
The variance of X is mathematically represented as
V(X) = [tex]E(X^2) - [E(X)]^2[/tex]
Substituting values
[tex]V[X] = 24 - 4^2[/tex]
[tex]V[X] =8[/tex]
The variance of Y is mathematically represented as
V(Y) = [tex]E(Y^2) - [E(Y)]^2[/tex]
Substituting values
[tex]V[Y] = 8 - 2^2[/tex]
[tex]V[X] =4[/tex]
The average of replacing the two component is
[tex]E(C) = 2 * E(X) + 4* E(Y)[/tex]
substituting value
[tex]E(C) = 50 + 2 * (4) + 4* (2)[/tex]
[tex]E(C) = 66[/tex]
The variance of replacing the two component is
[tex]V(C) = V(50 + 2X +4Y)[/tex] Note: The variance of constant is zero
and X and Y are independent
=> [tex]V(C) = 2^2 * V(X) + 4^2 * V(Y)[/tex]
substituting values
=> [tex]V(C) = 4 * 8 + 16 * 4[/tex]
=> [tex]V(C) = 32 + 64[/tex]
=> [tex]V(C) = 96[/tex]
The standard deviation is
[tex]\sigma = \sqrt{V(C)}[/tex]
substituting values
[tex]\sigma = \sqrt{96}[/tex]
[tex]\sigma = 9.798[/tex]
