B.Tech II Year I Semester Examinations, November - 2022

PROBABILITY AND STATISTICS

(Common to EEE, ECE, CSE, IT)

Time: 3 hours Max. Marks: 70

Note: 1. Question paper consists of Two parts (Part- A and Part- B)

2. Answer all the questions in Part-A and Any three questions from Part-B

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PART – A (10 * 2 = 20 Marks)

1. a) Define conditional probability.
2. b) If the probability density of a random variable is given by \( f(x) = \begin{cases} kx, & 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases} \) find the value of k.
3. c) Write any two properties of normal distribution.
4. d) Define point estimation and interval estimation.
5. e) Define null and alternative hypothesis.

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6. f) Write the formula for test statistic in testing the significance of difference between two means when samples are small.
7. g) Define queue length and waiting time in queuing theory.
8. h) Write the formula for probability distribution in (M/M/1: 8 / FIFO) queuing model.
9. i) Define stochastic matrix.
10. j) Write any two applications of Markov chains.

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PART – B (3 * 16 = 48 Marks)

1. a) State and prove Baye's theorem.
   b) A problem in mathematics is given to three students whose chances of solving it are 1/2, 1/3, and 1/4 respectively. What is the probability that the problem is solved?
2. a) A continuous random variable X has probability density function given by \(f(x) = \begin{cases} kx, & 0 \le x \le 2 \\ 2k, & 2 \le x \le 4 \\ 0, & \text{otherwise} \end{cases} \) Determine the value of k and calculate P(1= X =3).
   b) Fit a binomial distribution to the following data:
   \( \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline f & 2 & 14 & 20 & 34 & 22 & 8 \\ \hline \end{array} \)

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3. a) A random sample of size 100 has a mean of 15 and standard deviation of 3. Find the 95% confidence interval for the population mean.
   b) Explain the procedure of testing of hypothesis.
4. a) Two independent samples of sizes 9 and 8 from two normal populations have means 16, 13 and variances 9, 3 respectively. Find whether the means are significantly different.
   b) The following are the number of mistakes made in 5 successive days by 4 technicians working for a telephone company:
   \( \begin{array}{|c|c|c|c|c|} \hline \text{Technician} & 1 & 2 & 3 & 4 \\ \hline 1 & 2 & 3 & 4 & 5 \\ \hline 2 & 4 & 5 & 6 & 7 \\ \hline 3 & 6 & 7 & 8 & 9 \\ \hline 4 & 8 & 9 & 10 & 11 \\ \hline 5 & 10 & 11 & 12 & 13 \\ \hline \end{array} \)
   

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   Test whether the difference among the 4 sample means can be attributed to chance.
5. a) Explain the characteristics of queuing system.
   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:
   (i) The mean queue size (ii) The probability that the queue size exceeds 10.
6. a) Explain the applications of Markov chains.
   

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   b) The transition probability matrix for two states X1 and X2 is given by \( P = \begin{bmatrix} 0.4 & 0.6 \\ 0.2 & 0.8 \end{bmatrix} \). Initially, the system is in state X1. Determine:
   (i) The probability that it will be in state X2 after 3 steps.
   (ii) The steady state probabilities.

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