Saurastra University 2006 M.Sc Computer Science Stochastic Process & Econometrics) - Question Paper

M. Sc. (Part – II) exam
April / May – 2006
Statistics : Course No. – XII
(Stochastic Process & Econometrics)
Time : three Hours] [Total Marks : 100
Instructions : (1) Attempt any 5 questions, selecting at
lowest 2 from every part.
(2) every ques. carries equal marks.
part – A
1 (a) explain stochastic process. provide an example of it.
(b) In a random walk starting from origin, find the probability
of reaching to point ‘a’ before reaching to point ‘b’.
(c) find the expected duration of games played before it
terminate with the ruin of 1 of the player with p = q = 1
2.
2 (a) describe poisson process. explain the postulates of poisson
process.
(b) find the probability mass function of poisson processes.
(c) Show that sum of 2 independent poisson processes is a
poisson process.
3 (a) Consider a 2 state Markov chain P
a a
b b
=
-
-
LNM OQP 1
1
obtain
Pn and
n®¥
lim Pn .
(b) If P = LNM OQP 0 eight 0 2
0 three 0 7
. .
. .
, find Pn and
n®¥
lim Pn .
(c) Show that if in any game, bet is increased then probability
of ruin is reduced.
ST–2451] one [Contd...
4 (a) explain pure birth process. discuss assumptions of birth
process.
(b) find probability generating function of pure birth process.
(c) find the probability mass function of pure birth process.
5 (a) explain the probability of extinction.
(b) For a provided branching process starting with one member, show
that the probability of ultimate extinction is 1, if m £ one and
m > 1.
(c) Show that the ultimate probability of extinction is smallest
root of the formula s = pbsg.
part – B
6 (a) Consider the linear model eY, Xb, s2 Inj. explain different
assumptions. How will you estimate (i) ordinary lowest square
estimate b (ii) Var bbg.
(b) Consider a linear model Y = Xb + U; E Ubg= 0,
E UU In d 'i= s2 . Further suppose that there is a set of r
restrictions Rb = C where R and C (a vector) are known and
R is an r × k matrix or rank r. find lowest square estimate
of b under these constraints. Further under assumption of
normality, explain how you will test the hypothesis
H0 : Rb = C.
7 (a) discuss the issue of multicollinearity. explain its effect on
the efficiency of OLSE of b. How will you decide that the
multicollinearity exists in the data matrix ?
(b) define briefly how will you find Ridge estimator of b
when multicollinearity is current in the data matrix.
ST–2451] two [Contd...
8 (a) Consider the generalised linear model Y = Xb + U ; E Ubg= 0,
E UU' V d i= , V is known :
(i) find GLSE b of b.
(ii) Show that b is the BLUE of b.
(b) define Bartlett’s test to detect the heteroscedasticity of the
disturbance terms.
9 (a) define Durbin - Watson test and explain its usefulness.
(b) define 2 stage lowest squares estimator and show that it
is consistent under normal regularity conditions.
10 (a) find order and rank condition of identification in
simultaneous linear formula model.
(b) explain the identification of the subsequent 2 formula
model :
y1t 12 y2t u1t + b =
b21 y1t y2t v21 x1t u2t + + = .
ST–2451] three [ 100 ]

Earning:   Approval pending.