University Of Pune Question Paper
S.Y. B.A. Examination, 2010
MATHEMATICAL STATISTICS
General Paper – I
(Mathematical Statistics – II)
(New Course)
Time: 3 Hours Max. Marks: 80
N.B. : i) All questions are compulsory.
ii) Figures to the right indicate full marks.
iii) Use of calculator and statistical table is allowed.
iv) Symbols and abbreviations have their usual meaning.
v) Graph papers will be supplied on request.
1. Attempt any two of the following : (2×5=10)
a) i) Define probability mass function (p.m.f) of a random variable (r.v.) X.
ii) Determine ‘k’ such that the given function is a p.m.f :
[ ] , ,..... ; ( )!
21x
0k 1x
k xXp =
> − ==
b) Let X1 and X2 be independent Poisson random variables with parameters
M1 and M2 respectively. Obtain the distribution of Y = X1+X2.
c) With usual notations, describe a large sample test for testing Ho:μ =μ o against
H1:μ<μo. Use 5% level of significance (l.o.s.).
d) State any five advantages of chain base index number over fixed base index
number.
2. Attempt any four from the following : (4×5=20)
a) If X is a random variable (r.v.) with p.m.f. :
[ ]
..;
,,; ...
wo0
321x
x
6 xXp 22
=
= π ==
Check whether E(x) exists.
b) State and prove lack of memory property of geometric distribution. Also state
its interpretation.
P.T.O.
[3801] – 264 -2-
c) Customers arrive at a certain petrol pump in a certain city in a Poisson process
with an average time of 4 minutes. The time interval between services at the
petrol pump follows an exponential distribution.
The mean time taken to service one vehicle is one minute.
Find :
i) The probability that the petrol pump is idle.
ii) Expected length of the system.
iii) Average time spent by a customer in queue.
d) Discuss in detail income method of measuring national income.
e) A random sample of 200 bolts manufactured by machine ‘A’ and 100 bolts
manufactured by machine ‘B’ showed 20 and 5 defective bolts respectively.
Test whether machine ‘B’ is better than machine ‘A’ in respect of producing
defective bolts. Use 5% level of significance. (l.o.s.).
f) Show that X1.23 is uncorrelated with X2 and X3.
3. Attempt any three from the following : (3×10=30)
a) The joint p.m.f of (X, Y) is :
P(x,y)=q2py-2;X=1,2,...,(Y-1)
Y=2,3,4
0<P<1,q=1-p
Find
i) The marginal p.m.f of Y.
ii) The conditional distribution of X given Y = y
iii) E(X|Y=y)
iv) V(X|Y=y).
b) i) Let X1 and X2 are independent Poisson variates with parameters m1 and m2
respectively. Obtain the distribution of X1| X1+X2 = n ; n being nonnegative
integer. Identify the name of your distribution.
ii) Let X~NB (k,p). Obtain moment generating function of a r.v. X, hence
find mean and variance of it.
c) i) State the various components of time series. Discuss in detail two of them
with suitable example each.
ii) Describe moving average method to estimate trend in a time series.
d) Explain the concept of multiple correlation coefficient in case of trivariate
data. Also obtain an expression for multiple correlation coefficient R1.23 in
terms of total correlation coefficients r12, r13 and r23.
e) i) Explain the concept of ‘deflating’ of index numbers. Also state its uses.
ii) Discuss the census of products method of measuring national income.
4. Attempt any one from the following : (1×20=20)
a) i) Explain the term partial correlation coefficient in case of trivariate data.
Derive an expression for the partial correlation coefficient r12.3 in terms
of total correlation coefficients. 10
ii) Construct a cost of living index number for the following data and
interpret it. 4
Commodity Base year price Base year quantity Current year price
A 4 40 6
B 11 90 14
C 7 20 9
D 3 30 7
iii) Compute GNP, NNP at factor cost. Given that :
GNP at market price = 1,00,000 crores Rs.
Indirect taxes = 6,000 crores Rs.
Subsidies = 1900 crores Rs.
Depreciation = 8000 crores Rs.
Net income from abroad = 1800 crores Rs.
Also compute NDP.
Where GNP, NNP and NDP are Gross National Product, Net National
Product and Net Domestic Product respectively. 6
b) i) Define the following terms : 10
I) Statistical hypothesis
II) Critical region
III) Type I and Type II errors.
IV) Time series
V) Index numbers.
ii) Define the term ‘Cumulant Generating Function’ (C.G.F.). State and prove
its any two properties. 7
iii) Discuss additive model and multiplicative model in time series analysis. 3
______________
S.Y. B.A. Examination, 2010
MATHEMATICAL STATISTICS
General Paper – I
(Mathematical Statistics – II)
(New Course)
Time: 3 Hours Max. Marks: 80
N.B. : i) All questions are compulsory.
ii) Figures to the right indicate full marks.
iii) Use of calculator and statistical table is allowed.
iv) Symbols and abbreviations have their usual meaning.
v) Graph papers will be supplied on request.
1. Attempt any two of the following : (2×5=10)
a) i) Define probability mass function (p.m.f) of a random variable (r.v.) X.
ii) Determine ‘k’ such that the given function is a p.m.f :
[ ] , ,..... ; ( )!
21x
0k 1x
k xXp =
> − ==
b) Let X1 and X2 be independent Poisson random variables with parameters
M1 and M2 respectively. Obtain the distribution of Y = X1+X2.
c) With usual notations, describe a large sample test for testing Ho:μ =μ o against
H1:μ<μo. Use 5% level of significance (l.o.s.).
d) State any five advantages of chain base index number over fixed base index
number.
2. Attempt any four from the following : (4×5=20)
a) If X is a random variable (r.v.) with p.m.f. :
[ ]
..;
,,; ...
wo0
321x
x
6 xXp 22
=
= π ==
Check whether E(x) exists.
b) State and prove lack of memory property of geometric distribution. Also state
its interpretation.
P.T.O.
[3801] – 264 -2-
c) Customers arrive at a certain petrol pump in a certain city in a Poisson process
with an average time of 4 minutes. The time interval between services at the
petrol pump follows an exponential distribution.
The mean time taken to service one vehicle is one minute.
Find :
i) The probability that the petrol pump is idle.
ii) Expected length of the system.
iii) Average time spent by a customer in queue.
d) Discuss in detail income method of measuring national income.
e) A random sample of 200 bolts manufactured by machine ‘A’ and 100 bolts
manufactured by machine ‘B’ showed 20 and 5 defective bolts respectively.
Test whether machine ‘B’ is better than machine ‘A’ in respect of producing
defective bolts. Use 5% level of significance. (l.o.s.).
f) Show that X1.23 is uncorrelated with X2 and X3.
3. Attempt any three from the following : (3×10=30)
a) The joint p.m.f of (X, Y) is :
P(x,y)=q2py-2;X=1,2,...,(Y-1)
Y=2,3,4
0<P<1,q=1-p
Find
i) The marginal p.m.f of Y.
ii) The conditional distribution of X given Y = y
iii) E(X|Y=y)
iv) V(X|Y=y).
b) i) Let X1 and X2 are independent Poisson variates with parameters m1 and m2
respectively. Obtain the distribution of X1| X1+X2 = n ; n being nonnegative
integer. Identify the name of your distribution.
ii) Let X~NB (k,p). Obtain moment generating function of a r.v. X, hence
find mean and variance of it.
c) i) State the various components of time series. Discuss in detail two of them
with suitable example each.
ii) Describe moving average method to estimate trend in a time series.
d) Explain the concept of multiple correlation coefficient in case of trivariate
data. Also obtain an expression for multiple correlation coefficient R1.23 in
terms of total correlation coefficients r12, r13 and r23.
e) i) Explain the concept of ‘deflating’ of index numbers. Also state its uses.
ii) Discuss the census of products method of measuring national income.
4. Attempt any one from the following : (1×20=20)
a) i) Explain the term partial correlation coefficient in case of trivariate data.
Derive an expression for the partial correlation coefficient r12.3 in terms
of total correlation coefficients. 10
ii) Construct a cost of living index number for the following data and
interpret it. 4
Commodity Base year price Base year quantity Current year price
A 4 40 6
B 11 90 14
C 7 20 9
D 3 30 7
iii) Compute GNP, NNP at factor cost. Given that :
GNP at market price = 1,00,000 crores Rs.
Indirect taxes = 6,000 crores Rs.
Subsidies = 1900 crores Rs.
Depreciation = 8000 crores Rs.
Net income from abroad = 1800 crores Rs.
Also compute NDP.
Where GNP, NNP and NDP are Gross National Product, Net National
Product and Net Domestic Product respectively. 6
b) i) Define the following terms : 10
I) Statistical hypothesis
II) Critical region
III) Type I and Type II errors.
IV) Time series
V) Index numbers.
ii) Define the term ‘Cumulant Generating Function’ (C.G.F.). State and prove
its any two properties. 7
iii) Discuss additive model and multiplicative model in time series analysis. 3
______________
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