LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

NO 58

SECOND SEMESTER – APRIL 2008

ST 2805/ 2800 – PROBABILITY THEORY

 

 

 

Date : 06-05-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Section-A (10 × 2=20)

 

Answer ALL the questions.

∞

1.Show  that P( ∩  A_n  ) =1 if  P(A_n ) =1, n=1,2,3,…

n=1

2. Show that the limit of any convergent sequence of events is an event.
3. Define a random variable X and the σ- field induced by X.
4. Show that F(x) = P [X≤ x], x є R is continuous to the right.
5. Show that the probability distribution of a random variable is determined by its

distribution function.

6. Calculate E(X), if X has a distribution function F(x), where

F(x)  =   0         if x<0

x/2      if 0≤ x<1

1         if x ≥ 1.

7. If X₁ and X₂ are independent random variables and g₁ and g₂ are Borel functions, show

that g₁(X₁) and g₂(X₂ )are independent.

8. If Φ is the characteristic function (CF) of a random variable X, find the CF of (3X+4).
9. State Glivenko-Cantelli theorem.
10. State Lindeberg-Feller central limit theorem.

 

Section-B (5×8=40)

 

Answer any FIVE questions.

 

11. Explain the independence of two random variables X and Y. Is it true that if X and

Y are independent, X² and Y² are independent? What about the converse?

12. If X is a non-negative random variable, show that E(X) <∞ implies that
13. P(X > n) →0 as n→∞. Verify this result given that

f(x )= 2/ x³ ,    x ≥1.

13. In the usual notation, prove that

∞                                                           ∞

Σ   P [׀X׀ ≥ n]  ≤ E׀X׀ ≤    1 + Σ   P [׀X׀ ≥ n].

n=1                                                      n=1

 

14. Define convergence in probability. If X_n → X in probability, show that

X_n² + X_n  → X²+ X in probability.

15. If X_n → X in probability, show that X_n → X in distribution.
16. State and prove Kolmogorov zero-one law for a sequence of independent random

variables.

17. Using the central limit theorem for suitable Poisson random variables, prove that

n

lim   e^-n   Σ        n^k   = 1/2.

n→∞        k=0       k!

18. Find Var(Y), if the conditional characteristic function of Y given X=x is (1+(t²/x))^-1

and X has frequency function

f (x)  =1/x²,   for x ≥1

0,      otherwise.

 

Section-C (2×20= 40 marks)

 

Answer any TWO questions

 

19. (a) Define the distribution function of a random vector. Establish its

properties.                                           (8 marks)

(b) Show that the vector X =(X₁, X_2,…, X_p ) is a random vector if and only if X_j,

j =1, 2, 3… p is a real random variable.                 (8 marks)

(c)  If X is a random variable with continuous distribution function F, obtain the

probability distribution of F(X).                              (4 marks)

20 (a)  State and prove Borel zero –one law.

• If {X_n , n ≥1 }is a sequence of independent and identically distributed   random variables with common frequency function e^-x ,  x ≥ 0, prove that

P [lim (X _n / (log n)) >1] =1.                                        (12+8)

21  (a) State and prove Levy continuity theorem for a sequence of characteristic

functions.

• Let {X_n} be a sequence of normal variables with E (X_n) = 2 + (1/n) and

var(X) = 2 + (1/n²), n=1, 2, 3…Examine whether the sequence converges in distribution.                                                    (12+8 marks)

22 (a)  State and prove Kolmogorov three series theorem for almost sure convergence of

the series Σ X_n of independent random variables.                                    (12)

(b)  Show that convergence in quadratic mean implies convergence in probability.

Illustrate by an example that the converse is not true.    (8 marks)

 

 Go To Main page