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تحميل الدليل التدريبي

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Sample Exams of Stat 212

 

نماذج امتحانات مقرر 212 إحص

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

STAT 212  Midterm exam 1                                         2nd semester 2005

 

 

PROBLEM 1: Weather is observed for 40 days. The observer records whether it is clear (=0), cloudy(=1), rainy(=2). The following data was obtained :

  0  0  1  1  2  2  2  2  1  0  0  0  0  2  2  1  1  1  1  0  0  0  0  0  0  1  1 

  2  2  2  1  2  2  1  2  2  2  1  1  1

 1. Determine its state space S and parameter space T.

 2. If  Xn = type of weather on day n, determine the transition  probability matrix P.

 3. If it is known that the process starts in state 1, determine the probability  P(X2=2).

 

PROBLEM 2 : Two white balls and five green balls are placed in two boxes A and B so that box A contains 4 balls and box B contains 3 balls. At each step, a ball is selected at random from each box and the two balls are interchanged. Let Xn denotes the number of white balls in box A after the nth trial.

 1. Determine the transition probability matrix (TPM).

 2. If we start the experiment when box A has two(2) white balls, find the probability that :

            a) there is one white ball after 1 trial.

            b) there are two(2) white balls after 2 trials.

 

PROBLEM 3: Let { Xn , n=0,1,2,…} be a Markov chain with three states 0, 1, 2 and with transition probability matrix :

           Suppose that the initial distribution is P(X0=i) = 1/3 , i=0,1,2.

1. Find  P(X1 =1 \ X0  = 2) and P ( X2  = 2 \ X1  =1).

2. Find P(X0 = 2, X1  = 1, X2  =2),  P(X0  = 2, X1 = 1, X2 = 2, X3 = 1 ),

 P( X1 =1, X2 =1 , X3 =0 ), P( X1 = 1), and P(X2 = 2).

3. Find the stationary distribution of the Markov chain.

 

PROBLEM 4 :  Consider the Markov chain with state space S = { 1,2,…,6} and TPM :

                               0   1    0      0     0      0

                               0   0    1      0     0      0

                               1   0    0      0     0      0

                               0   0   0.2  0.5  0.3     0

                               0   0     0      0     0     1

                               0   0     0      0     1     0

 

1. Draw the graph of the Markov chain. Does state 2 communicate with state 5?.

    Is state 3 reachable from state 6 ?.

2. Find all classes of communicating states. Is the chain irreducible ?.

 

 

 

 

 

STAT 212  Midterm exam 1                                         2nd semester 2007

 

PROBLEM 1: Weather is observed for 40 days. The observer records whether it is clear (=0), cloudy(=1), rainy(=2). The following data was obtained :

 

  0  0  1  1  2  2  2  2  1  0  0  0  0  2  2  1  1  1  1  0  0  2  0  0  0  1  1 

  2  2  2  1  2  2  1  2  1  2  1  1  1

 

 1. Determine its state space S and parameter space T.

 2. If  Xn = type of weather on day n, determine the transition  probability matrix P.

 3. If it is known that the process starts in state 0, determine the probability  P(X2=2).

 

PROBLEM 2 : Three white balls and five green balls are placed in two boxes A and B so that box A contains 5 balls and box B contains 3 balls. At each step, a ball is selected at random from each box and the two balls are interchanged. Let Xn denotes the number of white balls in box A after the nth trial.

 

 1. Determine the transition probability matrix (TPM).

 2. If we start the experiment when box A has two(2) white balls, find the probability that :

            a) there are three (3) white balls  after 1 trial.

            b) there is one white ball after 2 trials

            c) Find  P(X3=4 / X1=2, X2=0) and P(X5=3 / X3=2).

 

PROBLEM 3: Let { Xn , n=0,1,2,…} be a Markov chain with three states 0, 1, 2 and with transition probability matrix :

 

 

1. Draw the graph of the Markov chain.

 Suppose that the initial distribution is P(X0=0) = 1/3,  P(X0=1) = 0, P(X0=2) = 2/3

2. Find  P(X1 =1 \ X0  = 2)   and   P ( X2  = 2 \ X1  =1).

3. Find   P(X0 = 2, X1  = 1, X2  =2),     P(X0  = 2, X1 = 1, X2 = 2, X3 = 1 ),

 P( X1 =1, X2 =1 , X3 =0 ), P( X1 = 1), and P(X2 = 2).

 

PROBLEM 4 :   A taxi moves between the airport (state 1), Hotel A (state 2), and Hotel B (state 3) according to a Markov chain with TPM :

                            

                                           

 

1. If the taxi is initially at the airport, what is the probability that it will be at Hotel A three moves later ?.

2. Suppose the taxi starts at the airport with probability 0.5 and starts at Hotel A and Hotel B with probability 0.25 each. What is the probability that the taxi will be at Hotel A three moves later ?.

STAT 212                                    Midterm  exam  # 2                2nd semester 2007

 

PROBLEM 1:  For the Markov chain with the following TPM :

 

 

1) Find all classes of communicating states and classify them as transient or recurrent.

2) Classify all states as periodic or aperiodic, finding the period of each state.

 

PROBLEM 2: Consider the TPM with state space S = { pizza 1, pizza 2 }  given by :  

 

                                  

1) Find the steady state probabilities of this Markov chain.

2)What proportion of customers, in the long run, purchase pizza 2 .

3) Determine the mean first time of return to pizza 1.

 

PROBLEM 3: Consider the  Markov chain with TPM and state space {0,1,2,3} given below:

 

                                                                 

 

1) Is  this an absorbing chain ?. Write the canonical form of this TPM. What are the matrices Q and  R?.

2) Find the fundamental matrix N = (I – Q)-1.

3) Starting  from state 1, determine the probability that the process is absorbed into i) state 1, ii) state

4)Determine the mean time to absorption starting respectively from state 0, and state 2.

 

PROBLEM 4 :  Consider the following TPM with state space {0, 1, 2} :

 

1)  Is the Markov chain irreducible ?. Determine steady state probabilities (π).

2) Find the expected return times  μi  , i=0,1,2  and the expected first time visit to state 2 starting from state 1, that is μ12.

STAT 212  Final exam                                                           2nd semester 2005

 

PROBLEM 1:  For the Markov chain with the following TPM :

 

 

1) Find all classes of communicating states and classify them as transient or recurrent.

2) Classify all states as periodic or aperiodic, finding the period of each state.

 

 

PROBLEM 2: Weather is observed for 40 days. The observer records whether it is clear (=0), cloudy(=1), rainy(=2). The following data was obtained :

  0  0  1  1  2  2  2  2  1  0  0  0  0  2  2  1  1  1  1  0  0  0  0  0  0  1  1 

  2  2  2  1  2  2  1  2  2  2  1  1  1

 1. Determine its state space S and parameter space T.

 2. If  Xn = type of weather on day n, determine the transition  probability matrix P.

 3. If it is known that the process starts in state 1, determine the probability  P(X2=2).

 

 

PROBLEM 3 : Two white balls and five green balls are placed in two boxes A and B so that box A contains 4 balls and box B contains 3 balls. At each step, a ball is selected at random from each box and the two balls are interchanged. Let Xn denotes the number of white balls in box A after the nth trial.

 1. Determine the transition probability matrix (TPM).

 2. If we start the experiment when box A has two(2) white balls, find the probability that :

            a) there is one white ball after 1 trial.

            b) there are two(2) white balls after 2 trials.

 

 

PROBLEM 4: Let { Xn , n=0,1,2,…} be a Markov chain with three states 0, 1, 2 and with transition probability matrix :

          

Suppose that the initial distribution is P(X0=i) = 1/3 , i=0,1,2.

1. Find  P(X1 =1 \ X0  = 2) and P ( X2  = 2 \ X1  =1).

2. Find P(X0 = 2, X1  = 1, X2  =2),  P( X1 = 1), and P(X2 = 2).

3. Find the stationary distribution π of the Markov chain.

 

 

 

 

 

PROBLEM 5 : Consider the TPM with state space S = { cola 1, cola 2 } of the Cola example given by :  

 

                                  

1) Find the steady state probabilities of this Markov chain.

2)What proportion of customers, in the long run, purchase cola 1 .

3) Determine the mean first time of return to cola 2.

 

PROBLEM 6: Consider the  Markov chain with TPM and state space {0,1,2,3} given below:

 

                                                                 

1) Is  this an absorbing chain ?. Write the canonical form of this TPM. What are the matrices Q and  R?.

2) Find the fundamental matrix N = (I – Q)-1.

3) Starting  from state 1, determine the probability that the process is absorbed into i) state 1, ii) state 3.

4)Determine the mean time to absorption starting respectively from state 0, and state 2.

 

PROBLEM 7 :  Consider the following TPM with state space {0, 1, 2} :

 

1)  Is the Markov chain irreducible ?. Determine steady state probabilities (π).

2) Find the expected return times  μi  , i=0,1,2  and the expected first time visit to state 2 starting from

state 1, that is μ12.

 

PROBLEM 8: Customers arrive at Adel's coffee shop according to a Poisson process ( X(t), t≥0) with rate 12 every hour.

1) What is the probability that exactly 2 customers will arrive to the coffee shop in the next 15 minutes?.

2) What is the probability that more than 3 customers will arrive during the next 30 minutes ?.

3) Find P(X(1) = 7 and X(3) = 15) and E(X(5)). Times 1, 3, and 5  are hours.

 

 

 

 

 

 

STAT 212  Final exam                                                          2nd semester 2007

 

PROBLEM 1:  For the Markov chain with the following TPM :

 

 

1) Find all classes of communicating states and classify them as transient or recurrent.

2) Classify all states as periodic or aperiodic, finding the period of each state.

 

 

PROBLEM 2:

 

PROBLEM 3 : Two white balls and five green balls are placed in two boxes A and B so that box A contains 4 balls and box B contains 3 balls. At each step, a ball is selected at random from each box and the two balls are interchanged. Let Xn denotes the number of white balls in box A after the nth trial.

 1. Determine the transition probability matrix (TPM).

 2. If we start the experiment when box A has two(2) white balls, find the probability that :

            a) there is one white ball after 1 trial.

            b) there are two(2) white balls after 2 trials.

 

 

PROBLEM 4: Let { Xn , n=0,1,2,…} be a Markov chain with three states 0, 1, 2 and with transition probability matrix :

          

Suppose that the initial distribution is P(X0=i) = 1/3 , i=0,1,2.

1. Find  P(X1 =1 \ X0  = 2) and P ( X2  = 2 \ X1  =1).

2. Find P(X0 = 2, X1  = 1, X2  =2),  P( X1 = 1), and P(X2 = 2).

3. Find the stationary distribution π of the Markov chain.

 

PROBLEM 6: Consider the  Markov chain with TPM and state space {0,1,2,3} given below:

 

                                                     

1) Is  this an absorbing chain ?. Explain why ?. Write the canonical form of this TPM.

    What are the matrices Q and  R?.

2) Find the fundamental matrix N = (I – Q)-1.

3) Starting  from state 1, determine the probability that the process is absorbed into i) state 2, ii) state 3.

4)Determine the mean time to absorption starting respectively from state 0, and state 1.

 

PROBLEM 7 :  Consider the following TPM with state space {0, 1, 2} :

 

1)  Is the Markov chain irreducible ?. Determine steady state probabilities ).

2) Find the expected return times  μi  , i=0,1,2  and the expected first time visit to state 2 starting from

state 1, that is μ12

ترجمة السؤال 2

 

 

 
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