LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

YB 39

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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