Andhra University 2005 B.Tech Information Technology - Probability, Statistics & Queueing Theory - Question Paper

MODEL PAPER

B.Tech (IT) Degree exam

Second Year - 1st Semester

PROBABILITY, STATISTICS & QUEUEING THEORY

Time: three hrs
Max Marks: 70

First ques. is Compulsory

ans any 4 from the remaining ques.

All ques. carry equal marks

ans all parts of any ques. at 1 place

1. ans the subsequent
i. State the axioms of probability.
ii. discuss confident intervals in estimation.
iii. discuss the method of lowest squares.
iv. What is rank correlation?
v. discuss kind I and II errors.
vi. State the properties of Regression coefficient?
vii. discuss conditional probability.

2. a) State and prove Baye’s formula on conditional probability.

b) We are provided 3 urns as follows:
Urn A contains three red and five white marbles
Urn B contains two red and one white marble
Urn C contains two red and two white marbles.
An urn is opted at random and a marble is drawn from the urn. If the Marble is red, what is the probability that it came from urn A?

3. a) Derive the Recurrence relation for finding the moments of a Binomial distribution.

b) If (x1, x2)= four x1 x2 e-(x12 x22) x1, x2>0
= 0 Otherwise.
obtain the marginal distributions of x1 and x2.4. a) Show that a Poisson distribution is a limiting case of Binomial. Also Derive the Moment generating function of a Poisson random variable.

b) Probability of a vehicle having an accident at a particular intersection is 0.0001. Suppose that 10,000 Vehicles perday travel through this intersection. What is the Probability of no accidents occurring? What is the Probability of 2 or more accidents.

5. a) describe a Normal distribution.

b) State and prove the properties of a Normal distribution.

6. a) Derive normal equations to fit y = a + bx by the method of lowest squares.

b) Fit a lowest squares parabola having the form y = a + bx + cx-2 to the subsequent data:

X: 1.2 1.8 3.1 4.9 5.7 7.1 8.6 9.8
Y: 4.5 5.9 7.0 7.8 7.2 6.8 4.5 2.77. a) Show that the correlation coefficient lies ranging from x and y -1 and +1

b) compute the correlation coefficient ranging from x and y for the subsequent data.

X: 65 66 67 67 68 69 70 72
Y: 67 68 65 68 72 72 69 718. Arrivals at a telephone booth are considered to be Poisson with an avg. time of 12 min. ranging from 1 arrival and the next. The length of a phone call is presumed to be distributed exponentially with mean four min.

a) obtain the avg. number of persons waiting in the system.
b) What is the probability that a person arriving at the booth will have to wait in the queue?
c) What is the probability that it will take him more than 10 mm. altogether to wait for the phone and complete his call?
d) Estimates the fraction of the day when the phone will be in use.
e) The telephone department will install a 2nd booth, when convinced that an arrival has to wait on the avg. for at lowest three min. for phone. By how much the flow of arrivals should increase in order to justify a 2nd booth?

Earning:   Approval pending.