LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION – STATISTICS
SECOND SEMESTER – APRIL 2004
ST 2800/S 815 – PROBABILITY THEORY
02.04.2004 Max:100 marks
1.00 – 4.00
SECTION – A
Answer ALL questions (10 ´ 2 = 20 marks)
- Show that if = 1, n = 1, 2, 3, …
- Define a random variable and its probability distribution.
- Show that the probability distribution of a random variable is determined by its distribution function.
- Let F (x) = P (Prove that F (.) is continuous to the right.
- If X is a random variable with continuous distribution function F, obtain the probability distribution of F (X).
- If X is a random variable with P [examine whether E (X) exists.
- State Glivenko – Cantelli theorem.
- State Kolmogorov’s strong law of large number (SLLN).
- If f(t) is the characteristic function of a random variable, examine if f(2t). f(t/2) is a haracteristic function.
- Distinguish between the problem of law of large numbers and the central limit problem.
SECTION – B
Answer any FIVE questions (5 ´ 8 = 40 marks)
- The distribution function F of a random variable X is
0 if x < -1
F (x) = if -1 £ x < 0
if 0 £ x < 1
1 if 1 £ x
Find Var (X)
- If X is a non-negative random variable, show that E(X) < ¥ implies that
- P [ X > n] ® 0 as n ® ¥. Verify this result given that
f(x) = .
- State and prove Minkowski’s inequality.
- In the usual notation, prove that
.
- Define convergence in quadratic mean and convergence in probability. Show that the former implies the latter.
- Establish the following:
- If Xn ® X with probability one, show that Xn ® X in probability.
- Show that Xn ® X almost surely iff for every > 0, is zero.
- {Xn} is a sequence of independent random variables with common distribution function
0 if x < 1
F(x) =
1- if 1 £ x
Define Yn = min (X1, X2 , … , Xn) . Show that Yn converges almost surely to 1.
- State and prove Kolmogorov zero – one law.
SECTION – C
Answer any TWO questions (2 ´ 20 = 40 marks)
- a) Let F be the range of X. If and B FC imply that PX (B) = 0, Show that P can
be uniquely defined on (X), the s – field generated by X by the relation
PX (B) = P {X Î B}.
- b) Show that the random variable X is absolutely continuous, if its characteristic function f
is absolutely integrable over (- ¥, ¥ ). Find the density of X in terms of f.
- a) State and prove Borel – Zero one law.
- b) If {Xn, n ≥ 1} is a sequence of independent and identically distributed random
variables with common frequency function e-x, x ≥ 0, prove that
.
- a) State and prove Levy continuity theorem for a sequence of characteristic functions.
- b) Use Levy continuity theorem to verify whether the independent sequence {Xn}
converges in distribution to a random variable, where Xn for each n, is uniformly
distributed over (-n, n).
- a) Let {Xn} be a sequence of independent random variables with common frequency
function f(x) =, x ≥ 1. Show that does not coverage to zero with probability
one.
- b) If Xn and Yn are independent for each n, if Xn ® X, Yn ® Y, both in distribution, prove
that ® (X2 + Y2) in distribution.
- c) Using central limit theorem for suitable exponential random variables, prove that
.
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