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B.E/B.Tech. DEGREE EXAMINATION,MAY/JUNE 2007
Fourth Semester
Computer Science and Engineering
MA1252-PROBABILITY AND QUEUING THEORY
Answer all questions
(PART A-10*2=20marks)
1.A lot of integrated circuit chips consists of 10 good,4 with minor defects
and 2 with major defects. Two chips are randomly chosen from the lot. What
is the probability that atleast one chip is good.
2.A continuous random variable X has a p.d.f f(x)=k(1+x),2=x=5. Find
P(x<4).
3.One percent of jobs arriving at a computer system need to wait until
weekends for scheduling, owing to core-size limitations. Find the probability
that among a sample of 200 jobs, there are no jobs that have to wait until
weekends.
4.A fast food chain finds that the average time customers have to wait for
service is 45 seconds. If the waiting time can be treated as an exponential
random variable, what is the probability that a customer will have to wait
more than 5 minutes given that already he waited for 2 minutes?
5.The joint p.d.f of 2 random variables X and Y is f(x,y)=cx(x-y),o<2,-x
=0 , elsewhere.
Evaluate C.
6.Let (x,y) be a two dimensional random variable. Define covariance of
(x,y). If x and y are independent what will be the covariance of (x,y)?
7.Define a Markov chain and give example.
8.If the transaction probability matrix of a Markov Chain is
, find the limiting distribution of the chain.
9.In the usual notation of an M/M/1 queueing system, if ?=12 per hour and
µ=24 per hour find the average number of customers in the system.
10.Write Pollaczek-Khintchine formula and explain the notations
(Part B-5*16=80 marks)
11(a).(i).A binary communication channel carries data as one of 2 types of
signal denoted by 0 and 1. Due to noise, a transmitted 0 is sometimes
received as a 1 and a transmitted 1 is sometimes received as a 0. For a given
channel assume a probability of 0.94 that a transmitted 0 is correctly
received as a 0 and a probability of 0.91 that a transmitted 1 ia received as a
1. Further assume a probability of 0.45 of transmitted a 0. If a signal is sent
determine (1) a 1 is received (2) a 1 was transmitted given that 1 was
received (3) a 0 was transmitted given that a 0 was received (4) an error
occurs.
11(a).(ii).In a continuous distribution, the probability density is given by
f(x)=kx(2-x), 0<2.>
Or
11.(b)(i).The cumulative distribution function (c d f) of a random variable x
is given by
0 x<0
x2 0=x<1/2
F(x)=1-3/35(3-x)2 1/2=x<3
1 x=3
Find the p.d.f of x and evaluate P(1*1<1)>
11.(b)(ii).Find the moment generating function of the geometric random
variable with p.d.f. f(x)=pqy-1 , x=1,2,3…… and hence obtain its mean and
variance.
12(a).(i).The number of monthly break down of a computer is a poisson
distribution with mean equal to 1.8. Find the probability, that this computer
will function for a month (1) without a break down (2) with only one
breakdown and (3) with atleast one breakdown.
12(a).(ii).The lifetime x in hours of a component is modeled by a Weibull
distribution with ß=2. Starting with a large number of components, it is
observed that 15% of the components that have lasted 90 hours fail before
100 hours. Find the parameter a.
Or
12(b).(i).The marks obtained by a number of students in a certain subject are
approximately normally distributed with mean 65 and standard deviation 5.
If 3 students are selected at random from this group, what is the probability
that atleast one of them would have scored above 75? [Given that the area
between z=0 and z=2 under the standard normal curve is 0.4772].
12(b).(ii).Write the p.d.f of Gamma distribution. Find the MGF, mean and
variance.
13(a).(i).The joint density function of the random variable (x,y) is given by
f(x)= 8xy 0<1,0
0 elsewhere
Find the (1).Marginal density of y
(2).conditional density of x given y=y and
(3).P(x<1/2)
13(a).(11)Calculate the correlation coefficient for the following data:
x: 65 66 67 67 68 69 70 72
y: 67 68 65 68 72 72 69 71
or
13(b).(i).If x and y are independent exponential random variables each with
parameter 1, find the p.d.f of u=x-y.
13(b).(ii).State central limit Theorem in Lindberg-leby’s form. A random
sample of size 100 is taken from a population whose mean is 60 and
variance is 400. Using central limit theorem with what probability can we
assert that the mean of the sample will not differ from µ=60 by more than 4?
Area under the standard normal curve from 0 to 2 is 0.4772.
14.(a).(i).At the receiver of an AM radio, the received signal contains a
cosine carrier signal at the Carrier frequency ?0 with a random phase ?, that
is uniformly distributed over (0,2p). The received carrier signal is
x(t)=Acos(?0t + ?). Show that the process is second order stationary.
14.(a).(ii).Queries presented in a computer data base are following a Poisson
process of rate
?=6 queries per minute . An experiment consists of monitoring the data base
for ‘m’ minutes and recording N(m) the number of queries presented.
1).What is the probability that no queries in a one minute interval?
2).What is the probability that exactly 6 queries arriving in one minute
interval?
3).What is the probability of less than 3 queries arriving in a half minute
interval?
Or
14(b).(i).Assume that a computer system ia in any one of the three states:
busy, idle and under repair respectively, denoted by 0,1,2. Observing its
state at 2P.M each day, we get the transition probability matrix as p=
.Find out the 3rd step transition prob matrix. Determine the limiting
probabilities.
14(b).(ii).Obtain the steady state or long run probabilities for the population
size of a birth death process.
15(a).(i).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 distributed exponentially with mean 4 minutes.
(1).What is the average number of customers in the system?
(2).What fraction of the day the phone will be in use?
(3).What is the probability that an arriving customer have to wait?
15(a).(ii).There are three typists in an office. Each typist can type an average
of 6 letters per hour. If letters arrive for being typed at the rate of 15 letters
per hour.
(1).What is the probability that no letters are there in the system?
(2).What is the probability that all the typists are busy?
Or
15(b).(i).Explain an M/M/1, finite capacity queueing model and obtain
expressions for the steady state probabilities for the system size.
15(b).(ii).Patients arrive at a clinic according to Poisson distribution at a rate
of 30 patients per hour. The waiting room does not accommodate more than
14 patients. Examination time per patient is exponential with mean rate of 20
per hour.
(1).What is the probability that an arriving patient will not wait?
(2).What is the effective arrival rate?