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00Code No: R21052R10SET - 1 II B. Tech I Semester Supplementary Examinations Dec - 2013 PROBABILITY AND STATISTICS(Com. to CSE, IT)Time: 3 hoursMax. Marks: 75Answer any FIVE QuestionsAll Questions carry Equal Marks1.a) Define : i) Probabilityii) Conditional probabilityiii) Random experimentiv) Exhaustive eventsb) A big contains 10 white and 6 black balls. Four balls are successively drawn out and notreplaced. What is the probability that they are alternatively of different color?(8M+7M)22.a) A random variable X has probability density functionf(x)?bx?6x,0?x?1. Find theconstant ‘b’ such that P(x<b) = P(x> b).b) A random variable x has the following probability distribution.Values of x012345678P (x)a3a5a7a9a11a13a15a17ai) 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).(8M+7M)3.a) Define Binomial distribution. Show that the mean of the binomial distribution is the productof 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 |''|'||||''|''||'|'| 00Code No: R21052R10SET - 1 5.a) Test the significance of the difference between the means of the sample from the followingdata:Size of sampleMeanS.DSample A100614Sample B200636b) 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 ofblood pressure. Can it be concluded that the medicine will in general be accompanied byan 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.(8M+7M)8.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 |''|'||||''|''||'|'| 00Code No: R21052R10SET - 2 II B. Tech I Semester Supplementary Examinations Dec - 2013 PROBABILITY AND STATISTICS (Com. to CSE, IT) Time: 3 hoursMax. Marks: 75Answer any FIVE QuestionsAll Questions carry Equal Marks1. a) For any two events A and B, show thatP(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 theprobability that the sample is in the order( B G B G) ?(8M+7M) 12342. 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 berandom phenomenon, with a probability function specified by the function??x/f(x)?? ?Ae5,forx?0?0,otherwisei) 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)3. a) Define moment generating function. How is moment generating function used to obtainmoments.b) A die is thrown 8 times, find the probability that 3 will appear:i) exactly 2 timesii) at least 2 timesiii) 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? 22?? is an unbiased estimator of ? then show that?is a biased estimator of?(8M+7M)1 of 2 |''|'||||''|''||'|'| 00Code No: R21052R10SET - 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.x01234f214922031b) Define number of degrees of freedom. List the properties of t- distribution.(8M+7M)7.a) Write steps involved in constructing x chart.b) Write the formulae required to draw c-chart.(8M+7M)8.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 |''|'||||''|''||'|'| 00Code No: R21052R10SET - 3 II B. Tech I Semester Supplementary Examinations Dec - 2013 PROBABILITY AND STATISTICS(Com. to CSE, IT)Time: 3 hoursMax. Marks: 75Answer any FIVE QuestionsAll Questions carry Equal Marks1. a) If A and B are mutually exclusive events, show that P(A )p(A/A?B)?,P(A?B)?0P(A)?P(B)b) Two factories produce identical clocks. The production of the first factory consists of10,000 clocks of which 100 are defective. The second factory produces 20,000 clocks ofwhich 300 are defective. What is the probability that a particular defective clock wasproduced 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 Xx0123456f (x)k3k5k7k9k11k13ki) Find kii) Evaluate P(x<4), p (x?? 5) and p (3 < x?? 6)iii) What is the smallest value of x for whichp(X? x)?12?(7M+8M)3.a) Obtain the moment generating function of the random variable x having probability?x:0?x?1?density function:f(x)??2?x:1?x?2? ?0:elsewhereb) Show that the mean and variance are equal for Poisson distribution.(8M+7M)4.a) Samples of 5 measurements of the diameter of a sphere are recorded as 6.33, 6.37, 6.36, 6.32and 6.37cm. Under the assumption that the measured diameter is normally distributed.Find unbiased and efficient estimates ofi) True meanii) True variance.b) List the properties of estimators with the examples.(8M+7M)1 of 2 |''|'||||''|''||'|'| 00Code No: R21052R10SET - 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 boys543210No. of girls012345No. of families1456110884012Is this result consistent with the hypothesis that male and female births are equallyprobable?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 containedrespectively,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 and3defectives. If the fraction defective is said to be maintained at 0.02, construct a P-chart forthese 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?? .(8M+7M)2 of 2 |''|'||||''|''||'|'| 00Code No: R21052R10SET - 4 II B. Tech I Semester Supplementary Examinations Dec - 2013 PROBABILITY AND STATISTICS (Com. to CSE, IT) Time: 3 hoursMax. Marks: 75Answer any FIVE QuestionsAll Questions carry Equal Marks1.a) Explain the relative frequency definition and axiomatic definition of probability and list theaxioms of probability.b) State Baye’s theorem. Find the probability of drawing two red balls in succession from abag containing 3 red and 6 black balls when i) the ball that is drawn first is replaced2.ii) it is not replace.a) A random variable x has the following probability function value of x(8M+7M) x0123456p(x)k3k5k7k9k11k13ki) Find kii) Evaluate p ( x??4), p ( x ??5) andp( x ? x) ??12?x2 b) For the continuous probability functionf(x)?kxewherex?0Find: i) kii) meaniii) 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 themaximum error with 99% confidence.b) Ifx1,x2,???,xnis a random sample from a normal populationN(?, 1). Show that1n2t?∑x i,is an unbiased estimator of?2???1(8M+7M)n|''|'||||''|''||'|'|i??11 of 2 00Code No: R21052R10SET - 4 5.a) A machine puts out 21 defective articles in a sample of 500 articles. Another machine gives3 defective articles in a sample of 100. Are the two machines significantly different in theirperformance?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 natureof work. Test whether the nature of work is independent of the sex of the worker.SkilledSemi-SkilledMales4020Females1030b) Write a note on student’s t- distribution.(8M+7M)7.a) Define: statistical quality control. List down the advantages of statistical quality controlover 100% inspection process?b) In the average fraction defective of a large sample of products is 0.1537. Calculate thecontrol limits.(8M+7M)8.A telephone booth functions with Poisson arrivals spaced 10 min apart on the average andexponential 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 |''|'||||''|''||'|'| ................
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