M.Sc. (Semester II) (CBCS) Examination Nov/Dec-2018
Statistics
PROBABILITY THEORY
Time: 2½ Hours Max. Marks: 70
Instructions: Q.1 and Q.2 are compulsory.
Attempt any three questions from Q. 3 to 7.
Figures to the right indicate full marks.
Q.1 Choose the correct alternative: 05
A class is a collection of
numbers alphabets
sets none of these
A is said to be closed under complementation, if ∈ implies

∈ for all ∈ ∈
Both a and b None of these
If is the characteristic function of then 0
0 1
2 n
The sequence of sets where 3 −
1

converges t
0,3 0,3
0,3 0,3
The field generated by the intervals of the type −∞, ∈ is
called
Standard field Borel field
Closed field None of these
Fill in the blanks: 05
Simple function is linear combination of indicators of sets.
If is a probability measure on then
If then
The characteristic function of a random variable X is real, if and
only if
Probability is a normed and additive measure.
State whether following statements are true or false: 04
Union of finitely many fields is a field.
An increasing sequence of sets may or may not converge.
A null set may or may not be empty set.
Two mutually disjoint sets are always independent.
Q.2 Answer the following: 06
Prove or disprove: Mapping preserves all the set relations.
Define field and Also distinguish between them.
Write short notes on the following: 08
Conditional probability measure
Convergence in distribution
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Q.3 State and prove continuity property of probability measure. 07
Prove: Inverse image of a is also a 07
Q.4 Prove limit superior and limit inferior for a sequence of a sets. Find the
same for sequence where 1
1

5 −
1

∈
07
Prove or disprove: If X and Y are independent random variables, then
. .
07
Q.5 State and prove Fatou's lemma. 07
State and prove Monotone convergence theorem. 07
Q.6 Define convergence in distribution. Prove or disprove: Convergence in
probability implies convergence in distribution.
07
Prove or disprove: Convergence almost sure implies convergence in
probability.
07
Q.7 Define characteristic function and find the same for binomial random
variable.
07
Define Induced probability measure and show that it is a probability
measure.