LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 36

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2007

ST 2902 – PROBABILITY & STOCHASTIC PROCESSES

 

 

 

Date & Time: 26/04/2007 / 1:00 – 4:00      Dept. No.                                       Max. : 100 Marks

 

 

PART-A

Answer all the questions.                                                                              10×2=20 marks

 

1. Define sample space and give an example.
2. If A is a subset of B, show that P(A)≤P(B).
3. Define conditional probability.
4. When three events are said to be mutually independent?
5. Define normal distribution.
6. Provide two properties of a distribution function.
7. Define Markov process.
8. When a state of a Markov chain is called recurrent and transient?
9. Define excess life and current life of a renewal process.

10.Write any two applications of stochastic processes.

 

PART-B

Answer any two questions.                                                                             5×8=40 marks

 

11.If 12 fair coins are tossed simultaneously, find the probability of getting

(a) exactly 6 heads  (b) atleast 3 heads  (c) almost 10 heads (d) not more than 5 heads.

 

12.State and prove Boole’s inequality.

 

13.Find the mean and variance of the distribution that has the distribution function

F(x) = 0,          x<0,

= x/8,       0≤x<2,

=x²/16,     2 ≤x<4,

.                                =1,            4 ≤x.

14.Let X and Y have the joint p.d.f.

f(x,y)= 6x²y,    0<x<1,0<y<1,

= 0 ,        elsewhere.

Find (i) the marginals of X and Y.

(ii) P(0<X<3/4, 1/3<Y<2).

 

15.Let the random variables X and Y have the joint pdf

f(x,y) = x+y ,  0<x<1,0<y<1,

= 0,       elsewhere.

Find the correlation coefficient of X and Y.

 

 

16. Communication is an equivalence relation-Prove.

 

17.Determine the classes and the periodicity of the various states for a Markov chain with

transition probability matrix

 

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

the assumptions.

PART-C

Answer any two questions.                                                                          2×20=40 marks.

 

19. (a) Derive the Poisson process clearly stating the postulates.

(b)Explain different types of stochastic processes.

 

20. Let f(x₁,x₂)=21 x₁² x₂³ , 0<x₁<x₂<1,zero elsewhere,be the joint pdf of X₁ and

X₂.Find the conditional mean and variance of X₁ given X₂=x₂,  0<x₂<1

 

21.A Markov chain on states {0,1,2,3,4,5} has transition probability matrix

 

 

 

(a) Find all classes.

(b) Find periodicity of various states.

(c) Find recurrent states.

(d) Obtain mean recurrence time of states.

 

22.(a) Explain renewal  process in detail.

(b) Derive forward and backward Kolmogorov differential equations for a birth and

death,  process, clearly stating the postulates.

 

 

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