**Model papers of Andhra University **

**BTech Computer Science & Engineering **

**P**robability Statistics & Queueing Theory

**Second Year – First Semester**

Effective from the admitted batch of 2004-2005

Time: 3 hrs

Max Marks: 70

First Question is Compulsory

Answer any four from the remaining questions

All Questions carry equal marks

Answer all parts of any question at one place

1. Answer the following

i. State the axioms of probability.

ii. Explain confident intervals in estimation.

iii. Explain the method of least squares.

iv. What is rank correlation?

v. Explain Type I and II errors.

vi. State the properties of Regression coefficient?

vii. Explain conditional probability.

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

b) We are given three urns as follows:

Urn A contains 3 red and 5 white marbles

Urn B contains 2 red and 1 white marble

Urn C contains 2 red and 2 white marbles.

An urn is selected 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)= 4 x_{1}x_{2}e^{-(x12 x22)}x_{1}, x_{2}>0 = 0 Otherwise. Find the marginal distributions of x_{1}and x_{2}.

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 two or more accidents.

5. a) Define 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 least squares.

b) Fit a least squares parabola having the form y = a + bx + cx^{-2} to the following 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 2.7

7. a) Show that the correlation coefficient lies between x and y -1 and +1

b) Calculate the correlation coefficient between x and y for the following data.

X: 65 66 67 67 68 69 70 72 Y: 67 68 65 68 72 72 69 71

8. Arrivals at a telephone booth are considered to be Poisson with an average time of 12 min. between one arrival and the next. The length of a phone call is assumed to be distributed exponentially with mean 4 min.

a) Find the average 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 second booth, when convinced that an arrival has to wait on the average for at least 3 min. for phone. By how much the flow of arrivals should increase in order to justify a second booth?