Question

7. (+1) The cost C(Q) of producing a quantity Q of widgets to satisfy demand is C() 4000+20Q, but the quantity demanded is random. If the mean and standard deviation of demand are 500 and 200, respectively, then what are the mean and standard deviation of costs?

Answer 1

ANSWER:

MEAN OF Q = 500

STANDARD DEVIATION OF Q = 200

C(Q) = 4000 + 20 Q THIS IS IN THE FORM B + AX

E[X] =_{$\tiny \sum_{1}^{N} X(i) / N$}

MEAN = E[C(Q)] = E[4000 + 20Q] =4000 +E[20Q] ---AS FOR A CONSTANT EXPECTATION IS ITSELF

BY PROPERTIES OF EXPECTATION OR SUMMATION WE KNOW E[AX] = AE[X]

AS $\tiny \sum AX = A\sum X$     GIVEN IF A IS A CONSTANT

=> MEAN = 4000 + 20 E[Q] = 4000 + 20 * 500 = 14000

Varlax b] = a"Var(X You is because: Var aX+ b] = E[ (ax + b)2 ]-(E [ax + b)? - a E(x2)- a E (X) - a Var(X)

NOW STANDARD DEVIATION OF COST= sqrt of variance(C(Q)) = sqrt{ Var(4000 + 20Q)} = sqrt( 20² var(Q))

= 20 * sqrt(var(Q)

= 20 * SD (Q) = 20 * 200 = 4000