LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – STATISTICS
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THIRD SEMESTER – NOV 2006
ST 3809 – STOCHASTIC PROCESSES
(Also equivalent to ST 3806)
Date & Time : 27-10-2006/9.00-12.00 Dept. No. Max. : 100 Marks
Section-A (10 × 2=20 marks)
Answer ALL the questions
- Define (a) Stationary increments
(b) Independent increments of a stochastic process
- Define the period of a state of a Markov chain. Show that an absorbing state is recurrent.
- Let j be a state for which fjj(n) = n/(2(n+1)), n>0. Show that j is recurrent.
- Write down the postulates for a birth and death process.
- Define a Renewal process {N(t),t ≥ 0} and write down its renewal function.
- Define a submartingale.
∞
- Let {Xn, n≥0} be a Branching process with the off spring mean m<1. Evaluate E[ Σ Xn].
n=0
- Define a Brownian motion process.
- Show that a Markov Renewal process is a Markov Chain with one step transition probabilities.
- Distinguish between wide-sense and strictly stationary processes.
Section-B
Answer any FIVE questions (5× 8 = 40 marks)
- Show that a Markov chain is fully determined, when its initial distribution and the one step transition probabilities of the Markov chain are known.
- Define a transient state and prove that transience is a class property. For any state i and a transient state j, prove that
∞
Σ pij(n) <∞
n=1
- Show that in a two dimensional symmetric random walk, all the states are recurrent.
- Assume that a device fails when a cumulative effect of k shocks occur. If the shocks happen according to a Poisson process with the parameter λ, find the density function for the life T of the device.
- Obtain the system of differential equations satisfied by the transition probabilities of the Yule process and calculate its transition probabilities when the initial condition is
X(0) = N.
- Derive the integral equation satisfied by the renewal function of a Renewal process.
- Let {X(t) | t Є[0,∞)} be a standard Brownian motion process. Obtain the conditional distribution of X(t) given X(t1)= α and X(t2)=β, where t1<t<t2.
- If {Xn} is a Branching process and φ n (s) is the probability generating function of Xn, show that φ n satisfies the relation φ n (s)= φ n-k (φ k (s) ) for all k such that
k= 1,2,…,n.
Section-C
Answer any TWO questions (2×20 =40)
19.a. Define a recurrent state. (2 marks)
- State and prove the Chapman-Kolmogorov equations for a discrete time discrete space Markov Chain.(10 marks)
- Consider a random walk on the integers such that pi,i+1 = p, pi,i-1=q for all integers i (0<p<1,p+q=1). Determine p00(n).Also find the generating function of p00(n) .(8 marks)
20.a. Show that recurrence is a class property.(6 marks)
- Show that states belonging to the same class have the same period.(6 marks)
- If lm pjj(n)>0, show that j is positive recurrent and aperiodic.(8 marks)
n→∞
21.a Stating the postulates for a birth and death process, derive Kolmogorov backward differential equations.(2+6 marks)
- Obtain E[X(t)], where X(t) is a linear birth and death process.(12 marks)
22.a. Define a discrete time Martingale and show that the means of the marginal distributions are equal. (8 marks)
- State and prove the prediction theorem for minimum mean square error predictors.
(12 marks)
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