PROBLEM 18.2 Suppose that the duration of a phone call (in minutes) for a certai

Question

PROBLEM 18.2

Suppose that the duration of a phone call (in minutes) for a certain person can be modeled as an exponential distribution with parameter .  This person is trying to decide between two calling plans.  Let X = the call duration.  The first plan charges $0.10 per call plus $0.10 a minute and thus, on average, a phone call will cost E[0.10 + 0.10X].  The second plan charges $0.50 per call plus $0.06 per minute and thus, on average, a phone call will cost E[0.50 + 0.06X].

a. What is the expected cost of a phone call under the first plan?

Note that the answer will be an expression containing .

b. What is the expected cost of a phone call under the second plan?

c. is a parameter which can be any positive number.  For which values of is the first plan more cost efficient than the second plan.

NOTE:  the expected call duration is E[X] = 1/.

Explanation / Answer

Expected cost for I plan = E[0.10 + 0.10X].

= 0.1+0.1E(x) = 0.1+0.1 lemda

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Expected cost for II plan

E[0.50 + 0.06X] = 0.50+0.06E(x)

= 0.50+0.06 lemda

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I plan is more efficient when cost plan is lesser

i.e. when 0.1+0.1 lemda<0.50+0.06 lemda

i.e. when 0.04 lemda<0.4

Lemda < 10

Thus for positive lemda, when lemda <10 the I plan is more efficient as cost is lower

For lemda = 10 both plans have the same cost

For lemda >10, II plan is cheaper

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