LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 32

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2008

ST 1809 – MEASURE AND PROBABILITY THEORY

 

 

 

Date : 30-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION A

 

Answer ALL questions                                                                     2 *10 = 20

 

1. Show that  =
2. Define : Sigma field
3. Mention any two properties of set functions.
4. If   exists, then show that
5. Define Singular measure.
6. State the theorem of total probability.
7. If E (X^k) is finite, k > 0, then show that E (X^j) is finite for 0 < j < k.
8. Let X₁ and X₂­ be two iid random variables with pdf  . Find V(X₁ + X₂).
9. Show that E ( E ( Y | ġ ) ) = E (Y).
10. Define Convergence in r^th  mean of a sequence of random variables.

 

SECTION B

Answer any FIVE questions                                                                                    5 * 8 = 40.

 

1. Show that finite additivity of a set function need not imply countable additivity.
2. Consider the following distribution function.

If μ is a Lebesgue measure corresponding to F, compute the measure of

a.)          b.)

 

1. State and prove the Order Preservation Property of integral of Borel measurable functions.
2. Let μ be a measure and λ be a singed measure defined on the σ field  of subsets of Ω. Show that λ << μ   if and only if | λ | << μ.
3. State and prove Borel – Cantelli lemma.
4. Derive the defining equation of conditional expectation of a random variable given a σ field.

 

 

1. Let Y₁,Y₂,…,Y_n be n iid random variables from U(0,θ). Define

X_n = max (Y₁,Y₂,…,Y_n). Show that

 

1. State and prove the Weak law of large numbers.

 

SECTION C

Answer any TWO questions.                                                                         2 * 20 = 40

 

1.  a.) Show that every finite measure is a σ finite measure but the converse need not

be true.

b.) Let h be a Borel measurable function defined on. If   exists,

then show that  = , v  c € R.                                              (8+12)

1. a.) State and prove the extended monotone convergence theorem for a sequence

of Borel measurable functions.

b.) If X = (X₁,X₂,…,X_n) has a density f(.) and each X_i has a density f_i, i= 1,2,…,n ,

then show that X₁,X₂,…,X_n are independent if and only if

a.e. [λ] except possibly on the Borel set of Rⁿ with Lebesgue measure zero.

(12+8)

 

1. a.) If Z is ġ measurable and Y and YZ are integrable, then show that

E ( YZ | ġ ) = Z E (Y | ġ) a.e. [P].

b.) Show that  implies  but the converse is not true.

(10+10)

1.  a.) State and prove Levy inversion theorem.

b.) Using Central limit theorem for suitable Poisson variable, prove that

.                                                                           (12+8)

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