Question

A large stockpile of used pumps contains 20% that are in need of repair. A maintenance worker is sent to the stockpile with three repair kits. She selects pumps at random and tests them one at a time. If the pump works, she sets it aside for future use. However, if the pump does not work, she uses one of her repair kits on it. Suppose that it takes 10 minutes to test a pump that is in working condition and 30 minutes to test and repair a pump that does not work. Find the mean and variance of the total time i takes the maintenance worker to use her three repair kits.

 

Answer 1

Solution

Find the mean and variance of the total time

for  checking and 30minute to check and repaire. 

Let X be the number of pumps that work precede 3 defective pumps.

Then  \( X\sim Nb(3,0.2) \)

\( \implies E(X)=3\times \frac{0.8}{0.2}=12 \)

the total time : \( T(X)=10\times X+3\times 30=10X+90 \)

Then  \( E(T(X))=120+90=210 \)

\( V(T(X))=100\times V(X)=100\times 3\times \frac{0.8}{0.04}=6000 \)

 

Therefore. \( E(T(X))=210\hspace{2mm},V(T(X))=6000 \)