**Sample Exams of Stat 212**

** **

**نماذج امتحانات مقرر ****212**** إحص**

** **

__STAT 212 Midterm exam 1 2__^{nd} 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 X_{n }= 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**(X_{2}=2).

__PROBLEM 2 :__** **__Two__ white balls and __five__ __g__reen 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 X_{n} denotes the number of white balls in box A after the n*th* 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 { X_{n} , 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**(X_{0}=i) = 1/3 , i=0,1,2.

**1.** Find **P**(X_{1} =1 \ X_{0} = 2) and **P** ( X_{2} = 2 \ X_{1} =1).

**2.** Find** P**(X_{0} = 2, X_{1} = 1, X_{2} =2), **P**(X_{0} = 2, X_{1} = 1, X_{2} = 2, X_{3} = 1 ),

**P**( X_{1} =1, X_{2} =1 , X_{3} =0 ), **P**( X_{1} = 1), and **P**(X_{2} = 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 2__^{nd} 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 X_{n }= 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**(X_{2}=2).

__PROBLEM 2 :__** Three** white balls and **five** __g__reen 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 X_{n} denotes the number of white balls in box A after the n*th* 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(X_{3}=4 / X_{1}=2, X_{2}=0) and P(X_{5}=3 / X_{3}=2).

__PROBLEM 3:__ Let { X_{n} , 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**(X_{0}=0) = 1/3, **P**(X_{0}=1) = 0, **P**(X_{0}=2) = 2/3

**2.** Find **P**(X_{1} =1 \ X_{0} = 2) ** and** ** P** ( X_{2} = 2 \ X_{1} =1).

**3.** Find** P**(X_{0} = 2, X_{1} = 1, X_{2} =2), **P**(X_{0} = 2, X_{1} = 1, X_{2} = 2, X_{3} = 1 ),

**P**( X_{1} =1, X_{2} =1 , X_{3} =0 ), **P**( X_{1} = 1), and **P**(X_{2} = 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 2__^{nd} 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 μ_{i } , i=0,1,2_{ } and the expected first time visit to state 2 starting from state 1, that is μ_{12}._{ }

__STAT 212 Final exam 2__^{nd} 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 X_{n }= 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**(X_{2}=2).

__PROBLEM 3 :__** **__Two__ white balls and __five__ __g__reen 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 X_{n} denotes the number of white balls in box A after the n*th* 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 { X_{n} , 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**(X_{0}=i) = 1/3 , i=0,1,2.

**1.** Find **P**(X_{1} =1 \ X_{0} = 2) and **P** ( X_{2} = 2 \ X_{1} =1).

**2.** Find** P**(X_{0} = 2, X_{1} = 1, X_{2} =2), **P**( X_{1} = 1), and **P**(X_{2} = 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 μ_{i } , i=0,1,2_{ } and the expected first time visit to state 2 starting from

state 1, that is μ_{12}._{ }

_{ }

__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 2__^{nd} 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__ __g__reen 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 X_{n} denotes the number of white balls in box A after the n*th* 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 { X_{n} , 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**(X_{0}=i) = 1/3 , i=0,1,2.

**1.** Find **P**(X_{1} =1 \ X_{0} = 2) and **P** ( X_{2} = 2 \ X_{1} =1).

**2.** Find** P**(X_{0} = 2, X_{1} = 1, X_{2} =2), **P**( X_{1} = 1), and **P**(X_{2} = 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 **(π**).

2) Find the **expected return times** **μ**_{i } , i=0,1,2_{ } and the expected first time visit to state 2 starting from

state 1, that is** μ**_{12}

__ترجمة السؤال 2___{}

_{}

* *