CHRIST UNIVERSITY,BENGALURU - 560029
End Semester Examination March - 2016
Bachelor of Technology - IV Semester
Code: EC431
Subject: PROBABILITY AND QUEUING THEORY
Max.Marks: 100
Duration: 3Hrs
SECTION A
Answer All Questions
5X20=100
1
a) (i) A bag contains 40 tickets numbered 1, 2, 3, …, 40 of which four are drawn at random and (10)
arranged in ascending order (t1 < t2 < t3 <
t4 ). Find the probability of t3 being 25? ( 5 Marks)
(ii) A five figure number is formed by the digits 0, 1, 2, 3, 4 without repetition. Find the
probability that the number formed is divisible by 4. ( 5 Marks)
[OR]
2
b)
3
a) (i) The odds that a book will be reviewed favorably by 3 independent critics are 5 to 2, 4 to 3 (10)
and 3 to 4. Find the probability that
Find the moment generating function of
find the mean and variance
and hence
(10)
majority of the reviews will be favorable?
(ii) In a class 70% are boys and 30% are girls. 5% of boys, 3% of the girls are irregular to the
classes. What is the probability of a
student selected at random is irregular to the classes and what is the probability that the
irregular student is a girl?
[OR]
4
b) A continuous RV
5
a) i) Define a binomial distribution.
6
ii) A communication system consists of components, each of which will independently
function with probability The total system will be able to operate effectively if at least onehalf of its components function. For what values of is a 5-component system more likely to
operate effectively than a 3-component system.
[OR]
b) i) Show that an exponential distribution is a legitimate probability distribution.
(10)
7
ii) The mileage which car owners get with a certain kind of radial tyre is a RV having an
exponential distribution with mean 40,000 km. find the probabilities that these tyres will last
(i) at least 20,000 km (ii) at most 30,000 km.
a) i)Find the density function of Y=aX+b in terms of the density function of X.
(10)
has a pdf
Find
(10)
, mean and variance.
(10)
ii)Let X be a continuous random variable with probability density function
.Find the probability density function of
[OR]
8
b) If X and Y are independent random variables following
respectively. Find the value of
9
.
(10)
and
such that
a) In producing gallium arsenide microchips, it is known that the ratio between gallium and
arsenide is independent of producing a high percentage of workable wafer, which are main
components of microchips. Let X denote the ratio of gallium to arsenide and Y denote the
percentage of workable micro wafers retrieved during 1-hour period. X and Y are
independent random variable with joint density function is defined as
.
(10)
show that E(XY) = E(X)E(Y)
10
[OR]
b) If the pdf of a two dimensional random variable(X,Y) is given by
Find i) the value of k ii)
(iv) p(X<1/Y<3).
iii)
11
(10)
a) An electronic device consists of two components. Let X and Y (years) be the times to failure (10)
of the first and second components respectively. Assume that (X,Y) has the density
. Find the densities of the marginal distribution.
What is the probability that the first component will have a lifetime of 2 years or longer?
12
[OR]
b) Verify central limit theorem for the independent random variable
,where for each k,
(10)
.
13
a) i) If
(10)
is a wide-sense stationary process with autocorrelation
determine the second-order moment of the random variable
ii) Suppose that
is a process with mean
and autocorrelation
Determine the mean, variance of the RVs
and
.
[OR]
14
b) With usual notation prove that:
i) The sum of two independent Poisson process is a Poisson process.
ii) The difference of two independent two independent Poisson process is not a Poisson
process.
(10)
15
a) Two random process
(10)
and
and
If A and B are uncorrelated RVs with zero means and
the same variances and
i)verify whether
ii) prove that
16
17
are defined by
is a constant, then
and
and
are individually stationary in the wide sense or not.
are jointly wide sense stationary.
[OR]
b) A student’s study habits are as follows: If he studies one night, he is 70% sure not to Study the (10)
next night. On the other hand, if he does not study one night, he is 60% sure not to study the
next night as well. In the long run, how often does he study.
a) A supermarket has a single cashier. During peak hours, customers arrive at a rate of 20 per
hour. The average number of customers that can be processed by the cashier is 24 per hour.
Calculate
(10)
(i) The probability that the cashier is idle.
(ii) The average number of customers in the queuing system.
(iii) The average time a customer spends in the system.
(iv) The average number of customers in the queue.
18
19
[OR]
b) What is the importance of learning M/G/1 model in queuing theory? Also, mention its
application.
(10)
a) A travel center has three service counters to receive people who visit to book air tickets. The (10)
customers arrive in a poisson distribution with the average arrival of 100 persons in a 10-hour
service day. It has been estimated that the service time follows an exponential distribution.
The average service time is 15 minutes. Find the :
i.
ii.
iii.
iv.
20
Expected number of customers in the system
Expected number of customers in the queue
Expected time a customers in the system
Expected waiting time for a customer in the queue.
[OR]
b) Patients arrive at a doctor’s clinic according to poisson distribution at a rate of 30 patients per (10)
hour. The waiting room does not accommodate more than 9 patients.Examination time per
patient is exponential with mean rate of 20 per hour. Find the:
(i) Average number of patients in the clinic
(ii) Expected time a patient spends in the clinic.
(iii) Average number of patients in the queue.
(iv) Expected waiting time of a patient in the queue