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In Oak Hill, Florida, 90% of all sunny days are followed by sunny days while 80% of cloudy days are followed by cloudy days. A local hotdog vendor makes, on average, $300 on sunny days and $100 on cloudy days. Assume you want to build a Markov model that you can use to answer questions b., c., d., and e. below. a. Define the states of the system and give the one-step transition probability matrix. b. If today is sunny, what is the probability it is sunny 5 days from now

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Answer:

a)  Sunny     Cloudy

T = [  0.9        0.2  ]  Sunny

     [  0.1         0.8  ]  Cloudy

b) 0.72269

Step-by-step explanation:

Given that;

90% of all sunny days are followed by sunny days while 80% of cloudy days are followed by cloudy days;

i.e if today is sunny (90%), the next day is sunny so its 10% cloudy

and if today is cloudy (80%), the next day is cloudy so its 20% sunny

a)

Now let T be the transition probability matrix which is expressed as;

     Sunny     Cloudy

T = [  0.9        0.2  ]  Sunny

     [  0.1         0.8  ]  Cloudy

b)

If today is sunny, to find the probability that it will be sunny in 5 days from now = P^5.x where x = [ 1 ] sunny

                                     [ 0 ] cloudy

so

p^5X = P.P.P.P.P.x

P.x = [ 0.9    0.2 ] [ 1 ]  = [ 0.9 ]

        [ 0.1     0.8 ] [ 0 ]    [ 0.1 ]

(P^2).x = [ 0.9    0.2 ] [ 0.9 ]  = [ 0.83 ]

              [ 0.1     0.8 ] [ 0.1 ]      [ 0.17 ]

(P^3).x = p(p^2)x = [ 0.9    0.2 ] [ 0.83 ]  = [ 0.781 ]

                              [ 0.1     0.8 ] [ 0.17 ]      [ 0.219 ]

(P^4).x = p(p^3)x =  [ 0.9    0.2 ] [ 0.781 ]  = [ 0.7467 ]

                               [ 0.1     0.8 ] [ 0.219 ]    [ 0.2533 ]

(P^5).x = p(p^4)x =  [ 0.9    0.2 ] [ 0.7467 ]  = [ 0.72269 ]

                               [ 0.1     0.8 ] [ 0.2533 ]    [ 0.27731 ]

so (P^5).x = [ 0.72269 ] sunny

                   [ 0.27731 ]  cloudy

therefore the probability that it is sunny 5 days from is 0.72269

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