LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2006

                       ST 2902 – PROBABILITY THEORY AND STOCHASTIC PROCESSES

Date & Time : 28-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

PART – A

Answer ALL the questions                                                                                                              (10 ´ 2 = 20)

1. Define probability by classical method.
2. Give an example for a discrete probability distribution.
3. Define an induced probability space.
4. State the properties of a distribution function.
5. Define the distributed function of a continuous random variable.
6. Write the formula to find the conditional mean and variance of Y given X = x.
7. What do you mean by a Markov matrix? Give an example
8. Write a note on one-dimensional random walk.
9. Define (i) recurrence of a state           (ii) periodicity of a state

10. Define renewal function.

PART – B

Answer any FIVE questions.                                                                                                             (5 ´ 8 = 40)

11. State and prove Boole’s inequality.
12. Explain multinomial distribution with an example.
13. Given the dF

F(x) =       0     ,  x < – 1

=    ,  -1

=      1        ,  1

compute (a) P(-1/2 < X  1/2)         (b) P(X = 0)    (c) P(X = 1)       (d) P (2 < X  3).

14. Let X have the pdf f(x) = 2x,  0 < x < 1, zero elsewhere. Find the dF and p.d.f. of Y = X².

15. (a) When is a Markov process called a Markov chain?

(b) Show that communication is an equivalence relation.                                                              (2 + 6)

16. A Markov chain on states {0,1,2,3,4,5} has t.p.m.

Find the equivalence classes.

17. Find the periodicity of the various states for a Markov chain with t.p.m.

18. Derive the differential equations for a pure birth process clearly stating the postulates.

PART – C

Answer any TWO questions.                                                                                                           (2 ´ 20 = 40)

19. (a) The probabilities that the independent events A,B and C will occur are ¼, ½ , ¼ respectively.
    What is the probability that at least one of the three events will occur?

• Find the mean and variance of the distribution that has the dF

F(x)  =  0         ,  x < 0

=  x/8      ,  0  £  x < 2

=  x²/16   ,  2  £  x < 4

=  1         ,  4  £  x                                                                                                     (5 + 15)

20. If X₁ and X₂ have the joint p.d.f.

f(x₁,x₂₎ =

find     (i) marginal pdf of X₁ and X_2.

                                                (ii) conditional pdf  of X₂ given X₁ = x₁ and X₁ given X₂ = x₂.

(iii) find the conditional mean and variance of X₂ given X₁ = x₁ and

X₁ given X₂ = x_2.                                                                                                                  (4 + 4 + 12)                      

21. Derive a Poisson process clearly stating the postulates.

22. Derive the backward and forward Kolmogorov differential equations for a

birth and death process clearly stating the postulates.

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – April 2009

MT 2961 – PROBABILITY THEORY AND STOCHASTIC PROCESSES

Date & Time: 02/05/2009 / 1:00 – 4:00  Dept. No.                                                Max. : 100 Marks

PART-A

Answer all questions                                                                                      (10 x 2 = 20)

1. Find the value of k such that, 0<x<1, zero, elsewhere represents the pdf of a random variable.
2. Find the mgf of a random variable X with probability mass function, x = 1,2,3,…
3. If A and B are two events and show that .
4. Define convergence in distribution of a sequence of random variables.
5. Define a stochastic process. What are the different ways of classifying a stochastic process?
6. Let X and Y be two independent random variables with N(10,4) and N(15,5) respectively. What is the distribution of 2X+3Y?
7. State Bonferronni’s inequality.
8. Define the period of a state. When do you it is aperiodic?
9. Patients arrive at a clinic according to Poisson process with mean rate of 2 per minute. What is the probability that no customer will arrive during a 2 minute interval?
10. The joint pdf of two random variables is, 0<<<1, zero, elsewhere.  Obtain the conditional pdf of  given .

PART – B

Answer any five questions                                                     ( 5 x 8 = 40)

11. Show that the distribution function satisfies, , and right continuity.
12. Derive the mgf of the normal distribution. Hence obtain mean and variance.
13. Show that binomial distribution tends to Poisson distribution under some conditions, to be stated.
14. _{Let X have a pdf f(x) = , 0 < x < ∞ and Y be another independent random variable with pdf g(y) = ,0 < y < ∞ .obtain the pdf of U= .}
15. _{Let {} be a Markov chain with states 1,2 and 3 and transition probability matrix}

              _{If the initial distribution is (},,), find            

1. _i)
2.              ii)

iii)

16. _{Derive the expression for in a pure birth process with X(0) =0}
17. _{Let X and Y have the joint pdf,zero, elsewhere. Find E[Y|x].}
18. _{Prove that E(XY) = E(X) E(Y) when the random variables are continuous and independent. Is the converse true? Justify.}

_{PART – C}

_{Answer any two questions                                                 ( 2 x 20 = 40)}

19. a) State and prove Baye’s theorem.
20. b) Give an example to show that pair wise independence does not imply

independence of three events.

c). State and prove Boole’s Inequality.                                   ( 8 + 6 + 6)

20. a) State and prove Markov inequality. Deduce Chebyshev’s inequality.

b). Show that almost sure convergence implies convergence in        probability.  Is

the converse true?  Justify.

c). State and prove central limit theorem for a sequence of i.i.d        random variables.                                                ( 5 + 5 + 10 )

21. (a). State the postulates of Poisson process and derive an expression for .

(b). Obtain the pgf of a Poisson process. If  and  have

independent Poisson processes with parameters  and      respectively,

find.           ( 6 + 5+ 9)

22. a). Show that the Markov chain with the transition probability matrix

is ergodic.  Obtain the stationary distribution.

b). Show that communication is an equivalence relation.

c). Let  be the minimum obtained while throwing a die n-times.

Obtain the transition probability matrix.                         ( 12 + 4 + 4)

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – April 2009

ST 2902 – PROBABILITY THEORY AND STOCHASTIC PROCESSES

Date & Time: 02/05/2009 / 1:00 – 4:00  Dept. No.                                                 Max. : 100 Marks

PART-A

Answer all questions                                                                                       (10 x 2 = 20)                       

• Determine C such that_{, –∞ < x < ∞ is a pdf of a random variable X.}
• _{If the events A and B are independent show that and are independent.}
• _{Write the MGF of a Binomial distribution with parameters n and p. Hence or otherwise find}
• _{If two events A and B are such that , show that P(A) ≤ P(B)}
• _{Given the joint pdf of and, f(,) = 2, 0<<<1.obtain the conditional pdf of given .}
• _{Define a Markov process.}
• _{Define transcient state and recurrent state.}
• _{Suppose the customers arrive at a bank according to Poisson process with mean rate of 3 per minute. Find the probability of getting 4 customers in 2 minutes.}
• _{If has normal distribution N(25,4) and  has normal distribution N(30,9) and if  and  are independent find the distribution of 2 + 3.}
• _{Define a renewal process.}

_PART-B

_{Answer any five questions                                                          (5 x 8 = 40)}

• _{Derive the MGF of normal distribution.}
• _{Show that F (-∞) = 0, F (∞) = 1and F(x) is right continuous.}
• _{Show that binomial distribution tends to Poisson distribution under some conditions to be stated.}
• _{Let X and Y be random variables with joint pdf f(x,y) = x+y, 0<x<1, 0<y<1, zero elsewhere. Find the correlation coefficient between X and Y}
• _{Let {} be a Markov chain with states 1,2,3 and transition probability matrix}

              _{with, i = 1,2,3}

              _{Find i)}

1.                 ii)

• _{Obtain the expression for in a pure birth process.}
• _{State and prove Chapman-Kolmogorov equation on transition probability matrix.}
• _{Let X have a pdf f(x) = e}^-x_x^m-1_{, 0 < x < ∞ and Y be another independent random variable with pdf g(y) = e}^-y _y^n-1_,

     _{0 < y < ∞ .obtain the pdf of U= .}

_PART-C

_{Answer any two questions                                                        (2 x 20 = 40)}

• _{. a) State and prove Bayes theorem.}

1. _{b) Suppose all n men at a party throw their hats in the centre of the room.}        

             _{Each man then randomly selects a hat. Find the probability that none of  them will}

             _{get their own hat.                                          (10 + 10)}

• _{a) let {} be an increasing sequence of events. Show that}

          _{P(lim ) = limP(). Deduce the result for decreasing events.}

1. _{b) Each of four persons fires one shot at a target. let Ai , i = 1,2,3,4 denote}         

          _{the event that the target is hit by person i.  If Ai are independent and}

          _{P() = P() = 0.7, P() =0.9, P() = 0.4. Compute the probability that}

1. _{All of them hit the target}
2. _{Exactly one hit the target}
3. _{no one hits the target}
4. _{atleast one hits the target. (12 + 8)}

• _{. a) Derive the expression for in a Poisson process.}

1. _{b) If and  have independent Poisson process with parameters  and.}

           _{Obtain the distribution of  = γ given  +  = n.}

1. _{c) Explain Yules’s process.                                            (10 + 5 + 5)}

_{22). a) verify whether the following Markov chain is irreducible, aperiodic and}                   

            _recurrent

           _{Obtain the stationary transition probabilities.}

1. _{b) State the postulates and derive the Kolmogorov forward differential}                       

              _{equations for a birth and death process.                            (10 +10)}

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