B.Tech II Year I Semester (R18) Supplementary Examinations November/December-2023

PROBABILITY AND STATISTICS

(Common to CSE, IT & CSE(DS))

Time: 3 Hours Max. Marks: 70

Note: 1. Question Paper consists of two parts (Part-A and Part-B)

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2. Answer ALL the question in Part-A

3. Answer any FOUR questions from Part-B

PART-A

1. a) If A and B are independent events, prove that A and B^c are also independent. (2M)

b) Define Mathematical expectation. (2M)

c) Write the properties of Normal distribution. (2M)

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d) Define critical region. (2M)

e) Write the applications of t-distribution. (2M)

f) Define transient and recurrent states of a Markov chain. (2M)

g) Define Kendall’s Notation. (2M)

PART-B

2. a) State and prove Baye’s theorem. (7M)

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b) In a bolt factory machines A, B, C manufacture 25, 35, 40 percent of the total of their output 5, 4, 2 percent are defective bolts. A bolt is drawn at random from the product and is found to be defective. What are the probabilities that it was manufactured by machines A, B and C? (7M)

3. a) A random variable X has the following probability function: (7M)

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

b) Find the moment generating function of the exponential distribution f(x)= 1c e^-x/c, x>0 and hence find its mean and variance. (7M)

4. a) Out of 800 families with 5 children each, how many would you expect to have (7M)

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i) 2 boys and 3 girls ii) at least one boy iii) no girl iv) all girls?

Assume equal probabilities for boys and girls.

b) Fit a Poisson distribution to the following data: (7M)

5. 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 the population mean and standard deviation, and mean and standard deviation of the sampling distribution of means. (7M)

b) A random sample of size 100 is taken from a population whose standard deviation is ?=5. Given that the interval (47.5, 52.5) is a 95% confidence interval for the mean of the population. Find the mean of the sample. (7M)

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6. a) Explain i) Null hypothesis ii) Alternative hypothesis iii) Type I and Type II errors (7M)

b) The mean life time of 200 bulbs of a company is found to be 2600 hours with a standard deviation of 20 hours. Test the hypothesis that the average lifetime of bulbs is 2590 hours at 1% level of significance. (7M)

7. a) A cubical die was thrown 9,000 times and 1, 2, 3, 4, 5, and 6 appeared with the following frequencies: (7M)

Calculate the value of ?2, and state whether the die is biased or not. (Table value of ?2 for 5 d.f. at 0.05 level is 11.070).

b) The following figures show the distribution of digits in numbers chosen at random from a telephone directory: (7M)

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Test whether the digits may be taken to occur equally often in the directory. (Given ?2_0.05 = 16.919 for 9 degrees of freedom.)

8. a) Define Stochastic matrix. Explain different types of stochastic matrices. (7M)

b) A training process is considered as a two-state Markov chain. If a trainee is in state I he passes the course in the first attempt with probability 0.8 or he is transferred to state II. If he is in state II, he passes the course in the first attempt with probability 0.6 or he remains in state II. In either state, a trainee cannot remain for more than three attempts. Draw the transition diagram. Find the transition probability matrix and calculate the probabilities of passing the course in one attempt, two attempts and three attempts respectively. (7M)

9. a) Explain (M/M/1): ( 8 / FIFO) Queuing Model. (7M)

b) In a railway yard, goods trains arrive at the 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: (7M)

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i) The mean queue size (line length)

ii) The probability that the queue size exceeds 10.

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