B.Tech II Year I Semester Examinations, December - 2018

PROBABILITY AND STATISTICS

(Common to CE, EEE, ME, ECE, CSE, EIE, IT, MCT, ETM, MMT, AE, MIE, PTM)

Time: 3 hours Max. Marks: 75

Note: This question paper contains two parts A and B.

Part A is compulsory which carries 25 marks. Answer all questions in Part A.

Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks.

PART - A (25 Marks)

1. 1. Define Random variable. [2M]

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   2. Find the moment generating function of Poisson distribution. [3M]
   3. Write the properties of Normal distribution. [2M]
   4. Define Null and Alternative hypothesis. [3M]
   5. Define Type I and Type II errors. [2M]
   6. Write the applications of t-distribution. [3M]

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   7. Write about the basic elements of queuing model. [2M]
   8. Explain transient and steady states. [3M]
   9. Define correlation and regression. [2M]
   10. Write the properties of regression coefficients. [3M]

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PART - B (50 Marks)

Unit - I

1. a) If X is a continuous random variable and y = aX + b, prove that E(y) = aE(X) + b and V(y) = a² V(x). [5M]
   b) A random variable X has the following probability function: [5M]
   

   x	0	1	2	3	4	5	6	7
   P(x)	0	K	2K	2K	3K	K2	2K2	7K2+K

   
   Determine (i) K (ii) Evaluate P(X < 6), (iii) Evaluate P(0 < X < 5)

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2. OR

   The diameter of an electric cable is assumed to be continuous variate with p.d.f f(x) = 6x(1-x), 0 ≤ x ≤ 1. Verify that above is p.d.f. Also find the mean and variance.

Unit - II

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1. a) Fit a Poisson distribution to the following data and test the goodness of fit. [5M]
   

   x	0	1	2	3	4
   f	192	100	24	3	1

   
   b) In a normal distribution, 7% of the items are under 35 and 89% are under 63. Determine the mean and variance of the distribution. [5M]



2. OR

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   If 20% of the bolts produced by a machine are defective, determine the probability that out of 4 bolts chosen at random (i) 1 (ii) 0 (iii) at most 2 bolts will be defective.

Unit - III

1. a) A population consists of the four numbers 3, 7, 11, 15. Consider all possible samples of size 2 which can be drawn without replacement from the population. Find [5M]
   i) The population mean ii) The population standard deviation iii) The mean of the sampling distribution of means. iv) The standard deviation of the sampling distribution of means.
   

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   b) Explain (i) Point estimation (ii) Interval estimation (iii) Bayesian estimation. [5M]



2. OR

   Samples of size 2 are taken from the population 1, 2, 3, 4, 5, 6 with replacement. Find
   i) The mean of the population. ii) The standard deviation of the population. iii) The mean of the sampling distribution of means. iv) The standard deviation of the sampling distribution of means.

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Unit - IV

1. a) A lady claims to be a tea taster. She names correctly 8 out of 10 cups of tea. Test whether she has any ability as a tea taster. [5M]
   b) A manufacturer claims that at least 95% of the equipment which he supplies to a factory conforms to specifications. An examination of a sample of 200 pieces of equipment reveals that 18 are faulty. Test his claim. [5M]



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2. OR

   Two horses A and B were tested according to the time (in seconds) to run a particular race with the following results:

   
   

   Horse A	28	30	32	33	33	29	34
   Horse B	29	30	30	24	27	29

   
   Test whether you can discriminate between the two horses.

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Unit - V

1. a) Obtain the equations of two lines of regression for the data. [5M]
   

   X	65	66	67	67	68	69	70	72
   Y	67	68	65	68	72	72	69	71

   
   b) In a railway marshalling yard, goods trains arrive at a rate of 30 trains per day. Assuming that the inter-arrival time follows an exponential distribution and the service time distribution is also exponential with an average of 36 minutes. Calculate the following: [5M]
   i) The mean queue size (line length) ii) The probability that the queue size exceeds 10.

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2. OR

   Calculate the coefficient of correlation and obtain the lines of regression for the following data:

   
   

   X	1	2	3	4	5	6	7
   Y	9	8	10	12	11	13	14

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