M.Sc. (Semester II) (CBCS) Examination Oct/Nov-2017
Statistics
PROBABILITY THEORY
Day Date: Friday, 17-11-2017 Max. Marks: 70
Time: 10:30 AM to 01.00 PM
Instructions: 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
Field is a class closed under
Complementation Finite union
Countable union Both and
If such that ∅ then the minimal field of contains sets.
2 3
4 8
If is a sequence of independent events, such that ∞ ∞



1 0
1 0
The characteristic function of a r. v. X is real, if and only if
is real valued is symmetric around zero
is measurable None of these
If Ω contains n elements, then the largest field of subsets of Ω
contains sets.

4
Fill in the blanks. 05
Simple function is linear combination of indicators of sets.
If is a probability measure on then Ω
If is the characteristic function of then
If is a convergent sequence of sets, then
If then set A is called as
State the following statements are True or False: 04
If and are two fields, then is also a field.
An increasing sequence of sets may or may not converge.
Any nonnegative r. v. can be expressed as a limit of sequence of
simple r. v. s.
If events B and C are mutually independent, then they are pairwise
independent also.
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SLR-MS-649
Q.2 Answer the following: 06
Show that mapping does not preserve set relations.
Define Borel field and show that −∞, ∈ is also a Borel set.
Write short notes on the following: 08
Generalised probability measure
Almost sure convergence
Q.3 Prove: If is a sequence of sets converging to set then
converges to .
07
Inverse image of a field is also a field. 07
Q.4 Define limit superior and limit inferior for a sequence of sets. Find the
same for sequence where 3,5 −
1

∈
07
Prove or disprove: If X and Y are independent random variables, then
. . .
07
Q.5 State and prove Monotone convergence theorem. 07
State and prove Borel- Cantelli lemma. 07
Q.6 Define convergence in probability. State and prove necessary and
sufficient condition for convergence in probability.
07
Prove or disprove: Almost sure convergence implies convergence in
probability.
07
Q.7 Define characteristic function and find the same for binomial random
variable.
07
Define expectation of simple as well as elementary random variable.