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 any five questions.
Q.1 and Q.2 are compulsory.
Attempt any three questions from Q. 3 to Q.7.
Figures to the right indicate full marks.
Q.1 Choose the correct alternative: 05
Which of the following statement is correct?
Every field is a -field
Union of fields is a field
Intersection of fields is a field
Ac} is a field, where A is non-empty subset of
For a sequence of sets
always exist always exist
always exist only and are true
A counting measure does not have
non-negativity normed
finite additivity -additivity
If X and Y are independent random variables

all the above
Characteristic function )of random variable X is
discrete continuous
may be or cannot be determined
Fill in the blank: 05
Σ
The minimal -field induced by an indicator function is

Let be a sequence of events such that
Then
Characteristic function is real if X is origin.
Finite additivity property of probability measure defined on
is
Page 2 of 2
The number of points in a set is called.
State true or false: 04
The generalized probability measure has a normed property.
If and are characteristic functions then
is
also a characteristic function.
If
→ then
→
Any polynomial of random variables is also a random variable.
Q.2 Define:
Field
-Field
Monotone Field
Give one example of each.
06
Write short notes on the following:
Probability measure
Strong Law of Large Numbers (SLLN)
08
Q.3 State and prove monotone property of probability measure. 06
Let Let A and B are subsets of
If then show that
If then show that
08
Q.4 Define a random variable. If X is random variable then prove that
is also a random variable.
07

Consider the function defined by
Where and are distinct. Obtain minimum -field induced
by X.
07
Q.5 Define almost sure convergence. Prove that almost sure
convergence implies convergence in probability.
06
Let
→ and
→ . Prove that

→

→
08
Q.6 State and prove dominated convergence theorem. 07
State Kolmogorov three series criterion for almost sure
convergence.
07
Q.7 State and prove Borel-Cantelli lemma. 07
Define characteristic function. Obtain characteristic function of
binomial distribution.