B.Tech II Year II Semester Examinations, May - 2019

PROBABILITY AND STATISTICS

(Common to CSE, IT)

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. Define the terms: Sample space, Event, and Random variable. (3M)

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2. A bag contains 4 white and 3 black balls. Two drawings of 2 balls are made without replacement. What is the probability that the first drawing will give 2 white balls and the second 2 black balls? (2M)
3. Define Marginal and Conditional probability density functions. (3M)
4. If X is a continuous random variable with probability density function given by \[ f(x) = \begin{cases} kx, & 0 \le x \le 2 \\ 2k, & 2 \le x \le 4 \\ -kx + 6k, & 4 \le x \le 6 \end{cases} \] find the value of k. (2M)
5. Define null hypothesis and alternative hypothesis. (3M)
6. Write about Type-I and Type-II errors. (2M)

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7. Write properties of good estimator. (3M)
8. If 300 screws are chosen at random from a large batch and 16 screws were defective, find 95% confidence interval for the proportion of defective screws in the batch. (2M)
9. Write about various types of control charts. (3M)
10. Explain about OC curve. (2M)

PART – B (50 Marks)

(Answer any one full question from each unit)

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UNIT – I

1. a) State and prove Baye’s theorem. (5M)
2. b) Two dice are thrown. Let X assign to each point (a, b) in S the maximum of its numbers i.e., X(a, b) = max(a, b). Determine the probability distribution of X. (5M)

   OR

3. a) In a certain town 40% have brown hair, 25% have brown eyes and 15% have both brown hair and brown eyes. A person is selected at random from the town.
   1. If he has brown hair, what is the probability that he has brown eyes also?
   2. If he has brown eyes, what is the probability that he has brown hair also?

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   3. What is the probability that he has either brown hair or brown eyes? (5M)
4. b) A random variable X has the following probability function:
   

   X	0	1	2	3	4	5	6	7
   P(x)	0	k	2k	2k	3k	k2	2k2	7k2+k

   
   Determine:
   1. k

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   2. Evaluate P(X<6), P(X≥6)
   3. P(0<X<5) (5M)

UNIT – II

1. a) Define Binomial distribution and derive mean and variance of Binomial distribution. (5M)
2. b) The diameter of an electric cable is assumed to be continuous random variable X with p.d.f f(x) = 6x(1-x), 0 ≤ x ≤ 1. Determine a number ‘b’ such that P(X < b) = P(X > b). (5M)

   OR

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3. a) Define Normal distribution and derive mean and variance of Normal distribution. (5M)
4. b) Out of 800 families with 5 children each, how many would you expect to have
   1. 3 boys
   2. at least one boy
   3. no girl

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   4. 2 or 3 girls (5M)

UNIT – III

1. a) Explain about tests of hypothesis concerning one mean. (5M)
2. b) The mean life time of 100 fluorescent light bulbs produced by a company is computed to be 1570 hours with a standard deviation of 120 hours. If µ is the mean lifetime of all the bulbs produced by the company, test the hypothesis µ=1600 hours, against the alternative hypothesis µ≠1600 hours, using a 5% level of significance. (5M)

   OR

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3. a) Explain about tests of hypothesis concerning one proportion. (5M)
4. b) Experience shows that 20% of a manufactured product is of top quality. In one day’s production of 400 articles only 70 are of top quality. Test the hypothesis at 0.05 level. (5M)

UNIT – IV

1. a) Explain the terms:
   1. Parameter
   2. Statistic
   3. Sampling distribution

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   4. Standard error (5M)
2. b) A random sample of 200 tins of coconut oil gave an average weight of 4.95 kgs with a standard deviation of 0.21 kgs. Find 95% and 99% confidence limits for the mean weight of the tins from which the sample was taken. (5M)

   OR

3. a) Write about estimation of one mean and one proportion. (5M)

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4. b) 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
   1. The population mean
   2. The population standard deviation
   3. The mean of the sampling distribution of means
   4. The standard deviation of the sampling distribution of means (5M)

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UNIT – V

1. a) Explain about control charts for measurements (X and R charts). (5M)
2. b) The following data give the number of defects in 10 machines: 3, 6, 4, 2, 5, 7, 8, 2, 5, 3. Draw a control chart for the number of defects. (5M) FirstRanker.com

   OR

3. a) Explain about control charts for attributes (p and c charts). (5M)
4. b) The following table gives the number of missing rivets noticed at final inspection of aircraft:
   

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   Aircraft number	1	2	3	4	5	6	7	8	9	10
   Number of missing rivets	4	12	6	8	10	3	15	7	0	9

   
   Draw a control chart for the number of defects and comment on the state of control of the process. (5M) FirstRanker.com