LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 31

SECOND SEMESTER – APRIL 2006

ST 2805 – PROBABILITY THEORY

(Also equivalent to ST 2800)

Date & Time : 24-04-2006/9.00-12.00 Dept. No. Max. : 100 Marks

Section-A

Answer ALL questions (10 ´ 2 = 20 marks)

With reference to tossing a regular die once and noting the outcome, identify completely all the elements of the probability space (Ω, A, P).
Show that the limit of any convergent sequences of events is an event.
Let F(x) = P [X < x], x Є R. Prove that F (.) is continuous to the left.
Write down any two properties of the distribution function of a random vector (X, Y).
If X is a random variable with P[X = (-1) ^k₂^k] = 1/2^k, k = 1, 2, 3…, examine whether E[X] exists.
If X²and Y² are independent, are X and Y independent?
Define almost sure convergence and convergence in probability for a sequence of random variables.
If Φ is the characteristic function (CF) of a random variable X, find the CF of (2X+3).
Let {Xn, n = 1, 2, …} be a sequence of independent and identically distributed (iid) n

N (μ, σ²) random variables. Define Yn = 1/n Σ X²_k, n = 1, 2, 3,… Examine K=1

whether Kolmogorov strong law of large numbers (SLLN) holds for
{Y_n, n = 1, 2, 3…}

Section – B

Answer any FIVE questions (5 × 8 = 40 marks)

Define the distribution function F(x) of a random variable X. State and establish its defining properties.
State and prove Minkowski’s inequality
State and prove Borel Zero- one law.
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

X is a random variable, which takes on positive integer values.

Define X_n = (n+1) if X=n

= n if X = (n+1)

= X otherwise

Show that X_n converges to X in quadratic mean.

Show that X_n → X in probability if and only if every subsequence of {X_n} contains a further subsequence, which converges almost surely.
Let {X_n, n ≥ 1} be a sequence of independent random variables such that X_n has uniform distribution on (-n, n). Examine whether the central limit theorem holds for the sequence {X_n, n ≥ 1}.

Section-C

Answer any TWO questions. (2 × 20 = 40 marks)

a. Define the probability distribution of a random variable. Show that the probability distribution of a random variable is determined by its distribution function.(8 marks)
Show that the vector X = (X₁, X₂, …, X_p) is a random vector if and only if X_j,

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

If X is a random variable with continuous distribution function F, obtain the probability distribution of F(X). (4 marks)

20.a. Show that convergence in quadratic mean implies convergence in probability. Illustrate by an example that the converse is not true. (8 marks)

21.a. State and prove Kolmogorov three series criterian for almost sure convergence of the series ∞

Σ X_n of independent random variables. (12 marks)

n=1

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

22.a. State and prove the continuity theorem for a sequence of characteristic functions.

(12 marks)

P [X_k = k^λ] = P [X_k = -k^λ] = 1/2, k = 1, 2, 3… Show that central limit theorem holds for

λ ≥ -1/2. (8 marks)

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