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)

 

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