Probability Statistics and Queuing Theory - Semester Paper

School of Computer Science, Engineering and Applications

BHARTHIDASAN UNIVERSITY

APRIL 2011 - IVth Semester

PROBABILITY, STATISTICS & QUEUEING THEORY

Date fo Exam : 09/05/12

INSTRUCTIONS :

Part - A : Answer all Questions

Part - B : Answer all Questions, either (a) or (b), one only

Part - C : Answer any Three only

PART - A

1. What is the probability that a Leap year selected at random contain 53 Saturadys or 53 Sundays ?

2. State the conditions to be satisfied by probabilty density function

3. If X is Binomially distributed with n = 6 such that 9P(x=4) = P(x=2). Find the mean and variance.

4. State any two properties of normal Distribution

5. State any two Properties of Correlation Coeefficent

6. From the lines of Regression 3x+2y=26 and 6x+y=31, determine the correlation coefficient

7. What is Type I Error and Type II Error

8. What is meant by critical region in hypothesis testing ?

9. What are transient and steady states in Queuing System

10. Give the forumula for the average number of Customers in the system and in the Queue for M/M/1 Model

PART - B

11. (a) An electronic assembly consists of two subsystems A and B. Given that P(A fails) = 0.15 , P(A & B fails) = 0.15 and P(B alone fails) = 0.15. Find i) P(A alone fails) ii) P(A fails / B has failed) [OR]

11. (b) Find the value K and mena value of the Random variables X, if its PDF is f(x) = Kx(1-x), 0=<x<=1

12. (a) i) State the Conditons for the Binomial distribution becomes Poisson bistribution

            ii) Write the PMF of poission distribution

            iii) State any two example situations for Poission [OR]

12. (b) Derieve the Momnet generating function of normal Distributions

13. (a) For the following calculate the rank correalation coefficient after assigining ranks

           X : 78 36 98 25 75 82 90 62 65 39

           Y : 84 51 91 60 68 62 86 58 63 47  [OR]

13. (b) Out of 1660 candidates who appeared for a competitive examination, 422 were successful. Out of these 256 had attended a coaching class and 150 of them came out successful . Find whether the Coaching was effictive as regards the success in the competitive examination 

14. (a) Describe the complete procedure for testing the hypothesis [OR]

14. (b) In a random sample 500, the mean is found to be 20. In another independent sample pf the size 400 the mean is 15. Could the samples have  been drawn from the same population with SD as 4 ?

15. (a) What are the characteristics of Queuing system [OR]

15. (b) Customers arrive at one man barber shop according to Poission process with the mean inter - arrival pattern of 12 minutes. Customers spend an average of 10 minutes with the barber's chair.

      i) What is the expected number of customers in the Barber shop?

      ii) What is the average time the customer spends in the queue ?

PART - C

16. If the Probability Mass Function of a random variable X is given by P(X=r) = K(r)cube where r = 1,2,3and4. Find the K, Mean, Variance and Distribution Function.

17. (a) Derive the Moment Generating function of Poission Distribution

      (b) In the distribution exactly normal 7% of the items are under 35 and 89% are under 63. What are the mean and standard deviation of the distribution

18. Determine the two lines of Regression and the Correlation Coeeficeint from the following data. Estimate the value of Y when X = 38 and X when Y = 18

      X : 22         26        29        30        31        31        34        35

      Y:  20         20        21        29        27        24        27        31 

19. Two independent samples of size 8 and 7 contaion the following values:

     S1: 19         17        15        21        16        18        16        14

     S2: 15         14        15        19        15        18        16        -

Is the difference between the sample means significant ?

20. A Self Service store employs one cashier at its counter. Nine Customers arrive on an average every 5 minutes while the cashier can serve ten customers in 5 minutes. Assembling Poission Arrival Rate and the Exponential Service Rate, Find :

(i) Average number of customers in the system

(ii) Average number of customers in the queue

(iii) Average tiem a customer spends in a system

(iv) Average time a customer waits before being served

TAGS : Sem 4, Semester Paper, Semester Question Paper, April 2012, Probability Statistics and Queuing Theory