Download 2013

Code No: R21052
R10
SET - 1
II B. Tech I Semester Supplementary Examinations Dec - 2013
PROBABILITY AND STATISTICS
(Com. to CSE, IT)
Time: 3 hours
Max. Marks: 75
Answer any FIVE Questions
All Questions carry Equal Marks
1.
a) Define : i) Probability
ii) Conditional probability
iii) Random experiment
iv) Exhaustive events
b) A big contains 10 white and 6 black balls. Four balls are successively drawn out and not
replaced. What is the probability that they are alternatively of different color? (8M+7M)
2.
a) A random variable X has probability density function f (x) bx 6x , 0 x 1 . Find the
constant ‘b’ such that P(x<b) = P(x> b).
b) A random variable x has the following probability distribution.
2
Values of x
0 1
2
3
4
P (x)
a 3a 5a 7a 9a
i) Determine the value of a.
ii) Find p (x<3), p (x 3) and P (0<x<5).
iii) Find the distribution function F(x).
5
11a
6
13a
7
15a
8
17a
(8M+7M)
3.
a) Define Binomial distribution. Show that the mean of the binomial distribution is the product
of the parameter P and the number of times n.
b) 1000 students have written an examination the mean of test is 35 and standard deviation is 5.
Assuming the distribution to be normal find:
i) How many students’ marks lay between 25 and 40? ii)
How many students get more than 40?
iii) How many students get below 20?
(7M+8M)
4.
a) Define population, sample and sampling. Give an example of each.
b) In a sample of size 5 results in the sample values of 8, 5, 9, 6 and 2. Find the sample mean
and sample variance.
(8M+7M)
1 of 2
|''|'||||''|''||'|'|
Code No: R21052
R10
SET - 1
5.
a) Test the significance of the difference between the means of the sample from the following
data:
Size of sample
Mean
S.D
Sample A
100
61
4
Sample B
200
63
6
b) Explain type I and type II errors in testing of hypothesis.
(8M+7M)
6.
a) A certain medicine is given to each of the 9 patients resulted in the following increase of
blood pressure. Can it be concluded that the medicine will in general be accompanied by
an increase in blood pressure using the data: 7, 3, -1, 4, -3, 5, 6, -4, -1.
b) Explain briefly the variance ratio test (or F-test).
(8M+7M)
7.
a) Write the expressions for central line and control limits for NP chart.
b) What is the use of control chart? Draw a typical control chart.
8.
(8M+7M)
a) Explain Queuing system. What is meant by transient and steady of a queuing system?
b) In a Railway marshalling yard, goods trains arrive at the rate of 30 trains / day. Assuming
that the inter-arrival time follows an exponential distribution and the service time (the time
taken to hump a train) distribution is also exponential with an average 30 min. Calculate:
i) Average number of trains in the queue.
ii) Probability that the Queue size exceeds 10.
(8M+7M)
2 of 2
|''|'||||''|''||'|'|
R10
Code No: R21052
SET - 2
II B. Tech I Semester Supplementary Examinations Dec - 2013
PROBABILITY AND STATISTICS
(Com. to CSE, IT)
Time: 3 hours
Max. Marks: 75
Answer any FIVE Questions
All Questions carry Equal Marks
1. a) For any two events A and B, show that P( A B) P ( A) P ( A B) P( A) P(B).
b) A bag contains 2 green and 3 black balls. A sample of size 4 is made. What is the
probability that the sample is in the order ( B G B G ) ?
(8M+7M)
1
2
3
4
2. a) Define: i) Probability density function.
ii) Probability mass function.
iii) Discrete random variable.
iv) Continuous random variable.
b) The length of time (in minutes) that a certain lady speaks on the telephone is found to be
random phenomenon, with a probability function specified by the function
 x/
Ae 5 , for x

0
0,
otherwise
i) Find the value of A that makes f(x) a probability density function.
ii) What is the probability that the number of minutes that she will take over the phone is,
more than 10 minutes?
(8M+7M)
f (x) 
3. a) Define moment generating function. How is moment generating function used to obtain
moments.
b) A die is thrown 8 times, find the probability that 3 will appear:
i) exactly 2 times
ii) at least 2 times
iii) at most once.
(8M+7M)
4. a) A sample of size 10 and standard distribution deviation 0.03 is taken from a population.
Find the maximum error with 99% confidence.
b) Define the problem of estimation. Show that if ˆ is an unbiased estimator of  then show
ˆ2
2
(8M+7M)
that  is a biased estimator of
1 of 2
|''|'||||''|''||'|'|
R10
Code No: R21052
SET - 2
5.
a) A college management claims that 80% of all single women appointed for teaching job get
married and quit the job within two years of time. Test this hypothesis at 5% level of
significance of among 200 such teachers. 112 got married within two years and quit their
jobs.
b) Write the test statistic:
i) in the test of significance for single mean.
ii) for the test of significance for single proportion and
iii) for the test of significance for difference of proportion.
(7M+8M)
6.
a) Fit a Poisson distribution to the following data and test the goodness of fit.
x
f
7.
8.
0
214
1
92
2
20
3
3
4
1
b) Define number of degrees of freedom. List the properties of t- distribution.
(8M+7M)
a) Write steps involved in constructing x chart.
b) Write the formulae required to draw c-chart.
(8M+7M)
a) Explain (m/m/1): ( / FCFS) Queuing model.
b) If for a period of 2 hours in a day (8-100m) trains arrive at the yard every 20 min, but the
service time continues to remain 36 min, then calculate for this period:
i) Probability that the yard is empty.
ii) Average queue length. On the assumption that the time capacity of the yard is limited to 4
trains only.
(7M+8M)
2 of 2
|''|'||||''|''||'|'|
R10
Code No: R21052
SET - 3
II B. Tech I Semester Supplementary Examinations Dec - 2013
PROBABILITY AND STATISTICS
(Com. to CSE, IT)
Time: 3 hours
Max. Marks: 75
Answer any FIVE Questions
All Questions carry Equal Marks
1.
a) If A and B are mutually exclusive events, show that
P( A )
p( A / A B) 
, P( A B) 0
P(A) P(B)
b) Two factories produce identical clocks. The production of the first factory consists of
10,000 clocks of which 100 are defective. The second factory produces 20,000 clocks of
which 300 are defective. What is the probability that a particular defective clock was
produced in the first factory?
(8M+7M)
2.
a) List the properties of probability distribution function.
b) A random variable x has the following function values of X
x 0 1
2
3
4
5
6
f (x) k 3k 5k 7k 9k 11k 13k
i) Find k
ii) Evaluate P(x<4), p (x 5) and p (3 < x 6)
iii) What is the smallest value of x for which p( X  x)  12 ?
3.
a) Obtain the moment generating function of the random variable x having probability
x
density function:
: 0  x 1
f (x)  2 x :1 x 
0 :elsewhere
2
b) Show that the mean and variance are equal for Poisson distribution.
4.
(7M+8M)
(8M+7M)
a) Samples of 5 measurements of the diameter of a sphere are recorded as 6.33, 6.37, 6.36, 6.32
and 6.37cm. Under the assumption that the measured diameter is normally distributed.
Find unbiased and efficient estimates of
i) True mean
ii) True variance.
b) List the properties of estimators with the examples.
(8M+7M)
1 of 2
|''|'||||''|''||'|'|
R10
Code No: R21052
SET - 3
5.
a) Two groups consisting of 400 and 500 persons have mean heights 68.5 inches and 66.1
inches and variances 6.4 and 6.0 respectively. Examine whether the difference in means of
the two groups?
b) Define: Statistical hypothesis. List the steps involved in the procedure of testing a
hypothesis.
(8M+7M)
6.
a) A survey of 320 families with 5 children each has the following distribution.
No. of boys
No. of girls
No. of families
5
0
14
4
1
56
3
2
110
2
3
88
1
4
40
0
5
12
Is this result consistent with the hypothesis that male and female births are equally
probable?
b) Write a short note on χ2- test for goodness of fit.
(8M+7M)
7.
a) Classify the control charts.
b) 35 - successive samples of 100 castings each taken from a production line contained
respectively, 3,3,5,3,5,0,3,2,3,5,6,5,9,1,2,4,5,2, 0,10,3,6,3,2,5,6,33,2,5, 1, 0,7,4 and 3
defectives. If the fraction defective is said to be maintained at 0.02, construct a P-chart for
these date.
(7M+8M)
8.
a) Describe the basic elements of a queuing system.
b) For the (m/m/1) queuing system. Find:
i) expected value of queue length n.
ii) Probability distribution of waiting time .
2 of 2
|''|'||||''|''||'|'|
(8M+7M)
R10
Code No: R21052
SET - 4
II B. Tech I Semester Supplementary Examinations Dec - 2013
PROBABILITY AND STATISTICS
(Com. to CSE, IT)
Time: 3 hours
Max. Marks: 75
Answer any FIVE Questions
All Questions carry Equal Marks
1.
a) Explain the relative frequency definition and axiomatic definition of probability and list the
axioms of probability.
b) State Baye’s theorem. Find the probability of drawing two red balls in succession from a
bag containing 3 red and 6 black balls when i) the ball that is drawn first is replaced
ii) it is not replace.
(8M+7M)
2.
a) A random variable x has the following probability function value of x
x
0 1
2
3
4
5
6
p(x) k 3k 5k 7k 9k 11k 13k
ii) Evaluate p ( x 4), p ( x 5) and
1
i) Find k
2
p( x  x)
2
x
b) For the continuous probability function f (x) kx e where x 0
Find: i) k
ii) mean
iii) variance
(8M+7M)
3.
a) Prove that the variance of binomial distribution is npq.
b) In a test on 2000 electric bulbs, it was found that bulbs of a particular make, was normally
distributed with an average life of 2040 hours and S.D of 60 hours. Estimate the number
of bulbs likely to burn for
i) More than 2150 hours,
ii) Less than 1950 hours
iii) More than 1950 but less than 2100 hours.
(8M+7M)
4.
a) A sample of size 10 and standard deviation 0.03 is taken from a population. Find the
maximum error with 99% confidence.
b) If x1, x2, , xn is a random sample from a normal population N (, 1) . Show that
1 n
t  ∑x i
2
, i timator of
s
a
n
u
n
b
i
a
s
e
d
e
s
2 1
(8M+7M)
n
i 1
1 of 2
|''|'||||''|''||'|'|
Code No: R21052
R10
SET - 4
5.
a) A machine puts out 21 defective articles in a sample of 500 articles. Another machine gives
3 defective articles in a sample of 100. Are the two machines significantly different in their
performance?
b) Differentiate two-tailed test of hypothesis from one-tailed test.
(8M+7M)
6.
a) The following table gives the classification of 100 workers according to sex and the nature
of work. Test whether the nature of work is independent of the sex of the worker.
Skilled
Semi-Skilled
Males
40
20
Females
10
30
b) Write a note on student’s t- distribution.
(8M+7M)
7.
a) Define: statistical quality control. List down the advantages of statistical quality control
over 100% inspection process?
b) In the average fraction defective of a large sample of products is 0.1537. Calculate the
control limits.
(8M+7M)
8.
A telephone booth functions with Poisson arrivals spaced 10 min apart on the average and
exponential call length averaging 3 min.
a) What is the probability that an arrival will have to wait more than 10 min before the phone
is free?
b) What is the probability that it will take a customer more than one 10 min altogether?
c) Estimate the fraction of a day that the phone will be in use.
d) Find the average no. of units in the system.
(15M)
2 of 2
|''|'||||''|''||'|'|