Original Question.
"Long distance calling plan A offers flat rate service at 10 cents per minute. Calling plan B charges 99 cents for every call 20 minutes; for calls over 20 minutes, the charge is 99 cents for the first 20 minutes plus 10 cents for every additional minute. (Note that hese plans measure your call duration exactly, without rounding to the next minute or even second. If your long-distance calls have exponential distribution with expected value t minutes, which plan offers a lower expected cost per call?"
from Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates , David J. Goodman
Second Meet Question
"Long-distance calling plan A offers rate service at 3 Baht per minute. Calling plan B changes 25 Baht for every call under 20 minutes; for calls over 20 minutes, the charge is 25 Baht for the first 20 minutes plus 3 Baht for every additional minutes (Note that these plans measure your call duration exactly, without rounding to the next minutes or even second.) If your long-distance calls have exponential distribution with expected value beta minutes, which plan offers a lower expected cost per call?"
The seconds times is on those pictures. The question is rather the same. It is changed only the numbers. and call extected value as beta.
In this solution I describe a 3 way to solve expected cost of plan B. E[B]
The First Solution is the cleverest method.
(the lower part of 1st page)
First , we observe the T variable that T is exponential then we find the probability for 0<=t<=20
P[0<=t<=20] = F(20)-F(0) = 1-e^(-20/beta) << Mass1
this is the first mass then the left mass is 1-(1-e^(-20/beta)) = e^(-20/beta) << Mass2
The part I call the clever is that this approach is use the property of Memoryless of exponential distribution
then we can cut out Many calculations.
the memoryless property lead to the equation at bottom of 1st page
the equation is E[B] = (The Mean position of Mass1)*(amount of Mass1) + (The Mean position of Mass2)*(amount of Mass2)
we left only the mean position of Mass2
from the memoryless property we got the mean of Mass2 is far from the starting of Mass2 as 3*beta (same as plan A) then we can conclude that the mean position of Mass2 is 25+beta (see graph 5th)
the left is the foundamental math to solved.
The Second Solution is the shortest method.
we know for E[B(t)] = integral ( B(t) f(t) dt )
then the after is straight forward. (the upper part of 2nd page)
The Third Solution is the straight method.
This method is just follow the step of other ordinary other questions.
(see the upper part of 1st page)
- First we consider the cost function.
- Second we consider the T random variable.
- Third we derive the B to the T random variable in CDF form.
then we got the mixed random variable. - Fourth change the CDF form to PDF form.
- Fifth we directly find the E[B] (see the lower part of 2nd page)
