R10Set No: 1
Code No: R31021
III B.Tech. I Semester Regular/Supplementary Examinations, January - 2014
COMPLEX VARIABLEDS AND STATISTICAL METHODS
(Electrical and Electronics Engineering)
Time: 3 HoursMax Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
**
1. (a) Show that real and imaginary parts of an analytic function are Harmonic.
(b) Show that (z) = xy + iy is everywhere continuous but not analytic.
2.(a) State and prove Taylor’s series.
(b) Evaluate ∫c(z) dz, where (z) =and c is the arc from z = -1 - i to z = 1 + i of
the cubical curve y = x3.
3.(a) Let a be an isolated singularity of (z) and if (z) is bounded on some neighborhood of a,
then, Prove that a is a removable singularity.
(b) Evaluatewhere c is the ellipse 4x2 + 9y2 = 9.
4. Find the transformation which maps the points z = 1, -i, -1 to the points w = i, 0, i respectively.
Also show that this transformation maps the region outside the circle z = 1 into the half plane
Real (w) ≥0.
5. (a) Two digits are selected at random from the digits 1 through 9.
i) If the sum is odd, what is the probability that 2 is one of the digit selected.
ii) If 2 is one of the digits selected, what is the probability that the sum is odd?
(b) Derive variance of the normal distribution.
6. (a) The mean voltage of a battery is 15 and standard deviation 0.2. Find the probability that
four such batteries connected in series will have a combined voltage of 60.8 or more volts.
(b) In a study conducted for 500 days, only on 4 days it was recorded that ‘lead’ content in a famous river exceeded 200mg/cm. Construct an upper 99% confidence limit for the probability that the ‘lead’ content in the river will exceed 200mg/cm on any one day.
7. (a) Discuss various types of Alternative hypothesis with a suitable example.
(b) An educator claims that average I.Q. of American college students is atmost 110, and that in a study made to test this claim 150 American College students, selected at random, had an average I.Q. of 111.2 with standard deviation of 7.2. Use a level of significance of 0.01 to test the claim of the educator.
8.(a) Explain, stating clearly the assumptions involved, the t-test for testing the significance of
the difference between the two sample means.
(b) The heights of six randomly chosen sailors are in inches: 63, 65, 68, 69, 71 and 72. Those
of 10 randomly chosen soldiers are 61, 62, 65, 66, 69, 69, 70, 71, 72 and 73. Discuss, the light
that these data throw on the suggestion that sailors are on the average taller than soldiers.
**
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R10Set No: 2
Code No: R31021
III B.Tech. I Semester Regular/Supplementary Examinations, January - 2014
COMPLEX VARIABLEDS AND STATISTICAL METHODS
(Electrical and Electronics Engineering)
Time: 3 HoursMax Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
**
1. (a) Show that every differentiable function is continuous, but converse is not true.
(b) Find the analytic function (z) = u + iw, where u = ex(x cosy - y siny).
2.(a) State and prove Cauchy integral formula.
(b) Find the Taylor’s series expansion of (z) = of convergence
3.(a) If (z) has a pole at z = a then prove that
(b) Evaluatewhere c is the circle z = 1.
about the point z = 0. Determine the region
4. If (z) is analytic function of z in a region R on the z - plane and let (z) 0 in R, then prove
that the mapping w = (z) is conformal for all points in R.
5. (a) Prove that Poisson distribution is the limiting case of Binomial distribution.
(b) Suppose three companies X, Y, Z produce T.V.’s. X produces twice as many as Y while Y
and Z produce the same number. It is known that 2% of X, 2% of Y and 4% of Z are defective.
All the T.V.’s produced are put into one shop and then one T.V. is chosen at random. Suppose
a T.V. chosen is defective, what is the probability that this T.V. is produced by company X?
6. (a) The mean of certain normal population is equal to the standard error of the mean of the
sample of 64 from that distribution. Find the probability that the mean of the sample size will
be negative.
(b) Why are interval estimates in most cases more useful than point estimates?
7. (a) Explain how a statistical hypothesis is tested?
(b) In a certain city 125 men in a sample of 500 were found to be smokers. In another city, the number of smokers was 375 in a random sample of 1000. Does this indicate that there is a greater population of smokers in the second city than in the first.
8. (a) Discuss test of independence of attributes with a suitable example.
(b) A random sample of 16 values from a normal population has a mean of 41.5 inches and sum of squares of deviations from the mean equal to 135 inches. Another sample of 20 values from an unknown population has a mean of 43.0 inches and sum of squares of deviations from their mean is equal to 171 inches. Show that the two samples may be regarded as coming from the same normal population.
**
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R10Set No: 3
Code No: R31021
III B.Tech. I Semester Regular/Supplementary Examinations, January - 2014
COMPLEX VARIABLEDS AND STATISTICAL METHODS
(Electrical and Electronics Engineering)
Time: 3 HoursMax Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
**
1. (a) Show that (z) = Pn(z) = a0 + a1z + a2z2 + a3z3 +….+an zn is continuous everywhere.
(b) Find the orthogonal trajectories of the family of curves r2cos2 = c = constant.
2.(a) State and prove Laurent series.
(b) Obtain the expansion for sinwhich is valid in 1 < z∞.
3.Evaluatep<1.
4.(a) Show that every bilinear transformation maps the circles in the z - plane onto the circles in
thew - plane.
(b) Determine the region of the w - plane into which the first quadrant of z - plane is mapped
by the transformation w = z2.
5.(a) A fair coin is tossed until a head or five tails occurs. Find the expected number of tosses of
the coin.
(b) A pair of dice is rolled 180 times. Determine the probability using normal distribution, that a total of 7 occur
i) atleast 25 times ii) between 33 and 41 times both inclusive iii) exactly 30 times.
6. (a) Explain the properties of a good estimator.
(b) Suppose the diameter of motor shafts in a lot have a mean of 0.249 inches and standard deviation 0.003 inches. The inner diameter of bearing in another lot has a mean of 0.25 inches and standard deviation of 0.002 inches. If a shaft and bearing are selected at random, find the probability that the shaft will not fit inside the bearing. Assume that both dimensions are normally distributed.
7. (a) Discuss critical region and level of significance of test of significance.
(b) A simple sample of the heights of 6400 Englishmen has a mean of 170 cm and a standard deviation of 6.4 cm, while a simple sample of heights of 1600 Austrians has a means of 172 cm and a standard deviation of 6.3 cm. Do the data indicate that the Austrians are on the average taller than the Englishmen?
8. (a) Write the properties of t-distribution.
(b) To test the effect of fertilizer on rice production 24 plots of land were chosen. Half of them
were treated with fertilizer and half without it. Mean yield on untreated plots was 4.8 quintals
with standard deviation of o.4 quintals, while on treated plots it was 5.1 quintals with standard
deviations of 0.36 quintals. Can we conclude that fertilizers provide significant improvement in
production?
**
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R10Set No: 4
Code No: R31021
III B.Tech. I Semester Regular/Supplementary Examinations, January - 2014
COMPLEX VARIABLEDS AND STATISTICAL METHODS
(Electrical and Electronics Engineering)
Time: 3 HoursMax Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
**
1. (a) State and prove necessary and sufficient condition for the function (z) to be analytic.
(b) Discuss the differentiability of the function (z) = z2 in the complex plane.
2.Find all possible Laurent series of (z) =about its singular points.
3.(a) Determine and classify the singularities of
(b) Evaluatewhere c is the simple closed path enclosing the points z = 0 and z = 1.
4. Find the bilinear transform which maps the points z = 0, -i, -1 into the points w = i, 1, 0. Find
the image of the line y = mx under this transformation.
5. (a) A distributor of bean seeds determines from extensive test that 10% of large batch of seeds
will not geminate. He sells the seeds in packets of 200 and guarantees 90% germination.
Determine the probability that a particular packet will violate the guarantee.
(b) Show that for normal distribution the quartile deviation, mean deviation and standard deviation are approximately 10:12:15.
6. (a) Derive an unbiased estimator of 2.
(b) A random sample of 500 pineapples was taken from a large consignment and 65 were
found to be bad. Show that the S.E. of the proportion of bad ones in a sample of this size is
0.015 and deduce that the percentage of bad pineapples in the consignment almost certainly lies between 8.5 and 17.5.
7.(a) What is meant by a statistical hypothesis? What are the two types of errors of decision that
arise in testing a hypothesis?
(b) A random sample of boots worn by 40 combat soldiers in a desert region showed an
average life of 1.08 years with a standard deviation of 0.05 years. Under standard conditions
the boots are known to have an average life of 1.28 years. Is there reason to assert at a level of
significance of 0.05 that use in the desert causes the mean life of such boots to decrease?
8. (a) Explain why the larger variance is placed in the numerator of the statistic F. Discuss the
application of F-test in testing if two variances are homogenous.
(b) Weekly sales (in Rs.) in small shops in 3 towns A, B, C are as follows
A = 620, 600, 740, 800
B = 410, 380, 350
C = 920, 870, 1040, 1030, 1010
Can we conclude that the shops in the 3 towns have the same average sales?
**
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