The federal government tries to boost small business activities by awarding annu

Question

The federal government tries to boost small business activities by awarding annual grants for projects. All bids are competitive but the chance of receiving a grant is highest if the owner has not received any during the last three years and lowest if awards were given in each of the last three years. Specifically, the probability of getting a grant if none were awarded in the last 3 years is 0.9. It decreases to 0.8 if one grant was awarded, 0.7 if two grants were awarded, and only 0.5 if three were received. (a) Describe this situation as a Markov chain by completing the 1-step transition matrix. Remember to clearly define your states. (b) Determine the expected number of awards per owner per year.

Explanation / Answer

none grant were awarded with probability 0.9

one grant was awarded with probability 0.8

two grants were awarded with probability 0.7

three grants were awarded with probability 0.5

one step transition matrix:

0 1 2 3

0[0.9 0.8 0.7 0.5]

expected number of awarded per owner per year i.e. E(x)=x.p(x)

x=0,1,2,3 p(x)= 0.9, 0.8, 0.7, 0.5

none grant were awarded with probability is 0.9= 0*09= 0

one grant was awarded with probability is 0.8= 1*0.8= 0.8

two grants were awarded with probability is 0.7= 2*0.7= 1.4

three grants were awarded with probability is 0.5= 3*0.5= 1.5

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