M.Sc.( Statistics) (Semester II) (CBCS) Examination, 2017
PROBABILITY THEORY
Day Date: Wednesday, 19-04-2017 Max. Marks: 70
Time: 10:30 AM to 01.00 PM
N.B. Attempt five questions.
Q. No. and Q. No are compulsory.
Attempt any three from Q. No. to Q. No.
Figures to the right indicate full marks.
Q.1 Choose correct alternative: 05
A Field is a class closed under
Complementation Finite intersection
Countable union All of the above
2 The minimal field of contains sets.
2 3 4 8
3 If is a sequence of independent events, such that
4 If is the characteristics function of r.v. then is
characteristic function of
None of these
5 The largest field of subsets of is called as
Power set Universal set
Sample class None of these
Fill in the blanks. 05
If is decreasing sequence of sets, then it converges to

Simple function is linear combination of indicators of
sets.
If A and B are disjoint sets, then
A random variable X is integrable, iff is integrable.
If and are independent random variables, then

Page 2 of 2
State the following sentence are True or False: 04
Every field is a field.
The indicator function of a random variable is a real valued
function.
A constant function is measurable with respect to any field.
Almost sure convergence implies convergence implies
convergence in distribution.
Q.2 Answer the following: 06
Define field and field. Distinguish between them.
Induced probability measure.
Write short notes on the following: 08
Generalised probability measure
Convergence in distribution
Q.3 Prove or disprove: Arbitrary intersection of fields is field. 07
Prove: If is non- descreasing sequence of sets converging to
set then converges to .
07
Q.4 Prove or disprove: Inverse mapping preserves all set relations. 07
Discuss: Measurable function ii) Borel function 07
Q.5 State and prove Monotone convergence theorem. 07
If is field of subsets of then prove that class B of all sets
whose inverse images belongs to is also a field.
07
Q.6 Define convergence in probability.
Define almost sure Convergence. Prove or disprove: Almost sure
convergence implies convergence in probability.
07
Q.7 Define indicator function and prove its any three properties. 07
Write short notes on: 07
Lebesgue measure
Lebesgue-Stieltj's measure