LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
SECOND SEMESTER – APRIL 2012
ST 2902 – PROBABILITY THEORY AND STOCHASTIC PROCESSES
Date : 23-04-2012 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
SECTION – A
Answer All the questions. (10 x 2 = 20 Marks)
- Let A and B two events on the sample space. If A C B, show that P (A) < P (B)
- If P (AÈB) = 0.7 , P(A) = 0.6 and P (B) = 0.5 , find P (AÇB) and P (AcÇB)
- Define conditional probability of events.
- Define normal distribution.
- Define a Renewal process.
- If X has the following probability distribution:
X = x: -3 -2 -1 0 1 2
P (X=x): 1/16 1/2 0 1/4 1/8 1/16
Find E (X).
- Let x be a nonnegative random variable of the continuous type with pdf f and let α>0. If Y = Xα , find the pdf of Y.
- Compute P (0 < X < 1/2 , 0 < Y < 1) if (X , Y) has the joint pdf
f (x , y) = x2 + xy/3 , 0 < x <1 , 0 < y < 2
0 , otherwise
- Define communication of states of a Markov chain.
- Write a note on Martingale.
SECTION – B
Answer any Five questions. (5 x 8 = 40 Marks)
- State and prove Boole’s inequality.
- A problem in statistics was given to 3 students and whose probabilities of solving it are respectively 1/2 , 3/4 and 1/4 . What is the probability that (i) at least one will solve
(ii) exactly two will solve (iii) all the three will solve if they try independently.
- If a random variable X has the pdf f (x) = 3x2 , 0 ≤ x < 1 , find a and b such that
(i) P (X ≤ a) = P (X >a) and (ii) P (X >b) = 0.05. Also compute P (1/4 < X < 1/2) .
- If X has pdf f (x) = k x2 e-x , 0 < x < ∞ , find (i) k (ii) mean (iii) variance
- Let X be a standard normal variable. Find the pdf of Y = X2.
- Explain the following (a) The Renewal function (b) Excess life (c) Current life
(d) Mean total life
- If f (x, y) = 6 x2 y , 0 < x < 1 , 0 < y < 1, find (i) P (0 < X < 3/4 Ç 1/3 < Y < 1/2)
(ii) (P (X < 1 | Y <2)
- (a) Prove that communication is an equivalence relation.
(b) Write the three basic properties of period of a state.
SECTION – C
Answer any Two questions. (2 x 20 = 40 Marks)
- (a) State and prove Bayes’ theorem.
(b) Consider 3 urns with the following composition of marbles.
Urn Composition of Marbles
White Red Black
I 5 4 3
II 4 6 5
III 6 5 4
The probabilities of drawing the urns are respectively 1/5, ¼ and 11/20. One urn was chosen at random and 3 marbles were chosen from it. They were found to be 2 red 1 black. What is the probability that the chosen marbles would have come from urn I, urn II or urn III?
- (a) If X has the probability mass function as P (X = x) = qxp ; x = 0, 1, 2, . . . . . ; 0 < p < 1 ,
q = 1-p find the MGF of X and hence find mean and variance.
(b) State and prove Lindeberg – Levy Central Limit Theorem.
- Let f (x1 y) = 8xy , 0 < x < y <1 ; 0 , elsewhere be the joint pdf of X and Y. Find the conditional mean and variance of X given Y = y , 0 < y < 1 and Y given x = x , 0< x < 1.
- (a) A Markov chain on states {0, 1, 2, 3, 4, 5} has transition probability matrix.
1 0 0 0 0 0
0 3/4 1/4 0 0 0
0 1/8 7/8 0 0 0
1/4 1/4 0 1/8 3/8 0
1/3 0 1/6 1/6 1/3 0
0 0 0 0 0 1
Find all equivalence classes and period of states. Also check for the recurrence of the
states.
(b) Derive Yule process assuming that X (0) = 1 (10 +10)
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