Question

Po, for n n! 1,2 k P) 4. (a) A bank manager at National Bank has compiled the following table of transition probabilities for the customer's use of various outlets of the bank in the University campus: Main branch Extension office ATM Outlet Main branch Extension office ATM 0.7 0.4 0.2 0.1 0.5 0.1 0.2 0.1 0.7 (i) Which outlet tends to have the highest "loyalty? Why? (ii) What proportion of customers who banked at the main branch this time can be expected to bank at the extension office two periods later? (ii) If the bank has 10,000 customers who use these outlets. In st eady state, how many of them would use each of these three outlets in a given period?

Answer 1

i) From the transition probability matrix, it is evident that the people visiting the main branch the current period will visit 70% of the time to the main branch in the next period and for the ATM also same applies. So they are having highest loyalty.

ii) two periods later probability

0.7 0.1 0.2 p0.4 0.5 0.1 0.2 0.1 0.7

=

0.57 0.14 0.29 0.5 0.3 0.2 0.32 0.14 0.54

So 14% of customer who serviced at the main branch after two periods may be serviced at extension branch.

c) Steady-state probability

$\pi1=0.7\pi1+0.4\pi2+0.2\pi3$

$\pi2=0.1\pi1+0.5\pi2+0.1\pi3$

$\pi3=0.2\pi1+0.1\pi2+0.7\pi3$

$\pi1+\pi2+\pi3=1$

Solving this

$\pi1=0.467, \pi2=0.167, \pi3=0.367$

so number of customers using main branch = 0.467*10000=4670

customers using the extension office = 0.167*10000=1670

customers using the ATM= 0.367*10000=3670

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