Please Help! 23 A car rental agency at the three New York airports finds the likelihood that cars rented andreturned to the various airports is given by the following table:Rented FromReturned ToProbabilityKennedyLaGuardiaNewark80%Kennedy10%10%KennedyLaGuardiaNewark30%50%20%LaGuardiaKennedyLaGuardiaNewark20%50%30%Newark Consider the car rental and return as a Markov chain with Kennedy as state 1, LaGuardiaas state 2, and Newark as state 3. What is the two-step transition matrix for this chain (thismeans two successive stages of car rentals)?056 029 0.150.33 0.44 0.230.37 0.41 0.220.69 0.18 0.1308 01 0.1a)0.35 0.42 0.23b)03 05 02c)0.40 0.38 0.2202 0.5 030.69 0.18, 0.130.7 02 0.108 a1 0.1d)0.43 0.38 0.19e)02 0.5 03f)02 05 030.37 0.42 0.2103 05 0203 05 02g) none of the others

Name: Please Help! 23 A car rental agency at the three New York airports fin
Uploaded: 2024-03-20T07:07:37.000Z
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Description: Please Help! 23 A car rental agency at the three New York airports finds the likelihood that cars rented andreturned to the various airports is given by the following table:Rented FromReturned ToProbabilityKennedyLaGuardiaNewark80%Kennedy10%10%Kenned

Introduction: Definition: Markov Chain A Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Use: They are stochastic processes for which the description of the present state fully captures all the information that could influence the future evolution of the process. Predicting traffic flows, communications networks, genetic issues, and queues are examples where Markov chains can be used to model performance. Data: Rented From Rented To Probability Kennedy Kennedy 80% LaGaurdia 10% Newark 10% LaGaurdia Kennedy 30% LaGaurdia 50% Newark 20% Newark Kennedy 20% LaGaurdia 50% Newark 30% Calculation Part: we first we have to convert percentages into individual values. Example: To convert X% into individual value is X%=X/100 based on above we have to convert given table of values. Rented From Rented To Probability Kennedy Kennedy 0.8 LaGaurdia 0.1 Newark 0.1 LaGaurdia Kennedy 0.3 LaGaurdia 0.5 Newark 0.2 Newark Kennedy 0.2 LaGaurdia 0.5 Newark 0.3 Based on question consider car rental and return as markov chain with kennedy as state 1, LaGaurdia as state 2 and Newark as state 3 is given by P= 0.8 0.1 0.1 0.3 0.5 0.2 0.2 0.5 0.3 To find two step transition matrix is denoted by P(2) and defined as P(2)=P*P so here we have perform 3x3 matrix multiplication to get two step transition matrix. P(2)=P*P = 0.8*0.8+0.1*0.3+0.1*0.2 0.8*0.1+0.1*0.5+0.1*0.5 0.8*0.1+0.1*0.2+0.1*0.3 0.3*0.8+0.5*0.3+0.2*0.2 0.3*0.1+0.5*0.5+0.2*0.5 0.3*0.1+0.5*0.2+0.2*0.3 0.2*0.8+0.5*0.3+0.3*0.2 0.2*0.1+0.5*0.5+0.3*0.5 0.2*0.1+0.5*0.2+0.3*0.3 = 0.64+0.03+0.02 0.08+0.05+0.05 0.08+0.02+0.03 0.24+0.15+0.04 0.03+0.25+0.10 0.03+0.10+0.06 0.16+0.15+0.06 0.02+0.25+0.15 0.02+0.10+0.09 = 0.69 0.18 0.13 0.43 0.38 0.19 0.37 0.42 0.21 P(2)=P*P 0.69 0.18 0.13 0.43 0.38 0.19 0.37 0.42 0.21 Result: The two step transition matrix P(2)=P*P is 0.69 0.18 0.13 0.43 0.38 0.19 0.37 0.42 0.21 So option d is correct.