# Andhra University BE/B.Tech Probability Statistics and Queuing Theory Previous Paper 2009

Question No.1 is compulsory. Answer any four from the rest.

Each question carries 14 marks.

1. (a) Explain sequence off Bernoulli trails.

(b) Explain negative binomial pmf.

(c) distinguish between increasing failure rate and decreasing failure rate.

(d) Explain moments.

(e) Explain random walk.

(f) Distinguish between discrete parameter and continuous parameter markove chains.

(g) Explain Goodness of fit tests.

2. (a) Prove that if two discrete random variables X any Y have the same PGF’s, then they must have the same distributions and Pmf’s.

(b) Explain Independent Random Variables, what is its use?

3. (a) Find the value of the constant K to that

Kx^{2}(1-x^{3}) 0<x<1

F(x)=

0 Other wise

(b) Explain Erlang and gamma distribution.

4. A certain telephone company charges for calls in the following way: $ 0.20 for the first three minutes or less; $ 0.08 per minute for any additional time. Thus if X is the duration of a call, the cost Y is given by:

0.20 0<=x<=3

F(x)= 0.20+0.08(x-3) x>=3

Find the expected value of the cost of a call (E[Y]), assuming that the duration of a call is exponentially distributed with a mean of 1/λ minutes. Use 1/ λ =1,2,3,4 and 5 minites.

5. (a) Explain Poisson process.

(b) Given that n>=1 arrivals have occurred in the interval [o,t], the conditional joint pdf of the arrival times T1T2………….Tn is given by

F[t1,t2,…….tn/N(t)=n]=n!/t^{n}

^{ }O<=t1<=t2<=……….<=tn<=t.

6. (a) Describe LRU-stack model.

(b) Describe M/M/1 Queue model.

7. (a) Explain tests on the populations mean.

(b) Explain maximum-likelihood estimation.

8. (a) Consider an arithmetic unit of a computer system with a modulo-m on-line fault-detector. As the modulus ma varies, the average detection latency y also varies. Given the following data, with two observations y for each value of m:

M_{i} : 3 5 7 11 13

Y_{i} : 1.45 1.30 1.20 1.10 1.05

(uuc)

1.5 1.26 1,23 1.08 1.03

Compute the coefficient of determination for the least-squares fit of the data.

(b) Explain Analysis of variance.

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