JNTU
JNTU B.Tech II PROBABILITY AND STATISTICS Previous Question Paper (May - 2019 - R15)
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 and may have a, b, c as sub questions.
Part A
Find the moment generating function of Normal distribution.
Answer
For a normal random variable X with mean μ and variance σ2, the probability density function is f(x) = (1 / (σ√(2π))) * exp(-(x - μ)2 / (2σ2)). The moment generating function MX(t) is defined as E[etX].
Calculate expectation and variance of X, if the probability distribution of the random variable X is given by:
| X | -1 | 0 | 1 | 2 | 3 |
| f | 0.3 | 0.1 | 0.1 | 0.3 | 0.2 |
Answer
The expectation E(X) is calculated as the sum of the products of each value of X and its corresponding probability f(x):
To calculate the variance V(X), we first find E(X2):
The variance is then given by the formula V(X) = E(X2) - [E(X)]2:
Show that the correlation coefficient is independent of change of origin and scale.
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The equations of two regression lines are 7x - 16y + 9 = 0 and 5y - 4x - 3 = 0. Find the means of x and y.
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It is desired to estimate the mean number of hours of continuous use until certain computer will first require repairs. If it can be assumed that σ = 48 hours, how large a sample be needed so that one will be able to assert with 90% confidence that the sample mean is off by at the most 10 hours.
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Define (i) Null Hypothesis (ii) Alternate Hypothesis (iii) Type I and Type II errors.
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Define the terms balking and reneging.
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Define: i) steady state ii) Transient state.
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Write the properties of a transition probability matrix.
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Is the matrix provided in the document irreducible?
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Part B
Derive mean and variance of Binomial distribution.
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Out of 800 families with 5 children each, how many would you expect to have: (i) 3 boys (ii) 5 girls (iii) either 2 or 3 boys? Assume equal probabilities for boys and girls.
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If the variance of a Poisson variate is 3, then find the probability that (i) x = 0, (ii) 0 < x ≤ 3 (iii) 1 ≤ x < 4.
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The marks obtained in statistics in a certain examination found to be normally distributed. If 15% of the students got ≥ 60 marks, 40% of the students got < 30 marks, find the mean and standard deviation.
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The random variables X and Y have the joint density function f(x, y) = c(2x + y) for 2 < x < 6, 0 < y < 5, else 0. Find: a) c, b) P(X > 3, Y > 2), c) P(X > 3), d) P(X + Y < 4), e) P(Y > 3).
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Find the coefficient of correlation and the two lines of regression between X and Y for the following data.
| X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Y | 12 | 11 | 13 | 15 | 14 | 17 | 16 | 19 | 18 |
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Two horses A and B were tested according to the time (in seconds) to run a particular track with the following results. Horse A: 28, 30, 32, 33, 33, 29, 34. Horse B: 29, 30, 30, 24, 27, 29. Test whether the two horses have the same running capacity.
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Two researchers adopted different sampling techniques while investigating some group of students to find the number of students falling into different intelligence level. Would you say that the sampling techniques adopted by the two researchers are significantly different using chi-square test?
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A TV repair man finds that the time spent on his job as an exponential distribution with mean of 30 minutes. If he repairs sets in the order in which they come in and if the arrival of the sets is approximately Poisson with an average rate of 10 per 8 hours a day. What is repair mans expected idle time in each day? How many jobs are ahead of the average set just bought in?
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A one person barber shop has six chairs to accommodate people waiting for haircut. Customers arrive at the average rate of 3 per hour and spend an average of 15 minutes for service. Find a) The probability that a customer can get directly into the barber chair upon arrival. b) Expected number of customers waiting for a haircut. c) Effective arrival rate. d) The time a customer can expect to spend in the barber shop.
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Explain classification of states and chains of markov process.
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A fair die is tossed repeatedly. If Xn denotes the maximum of the numbers occurring in the first n tosses, find the transition probability matrix P of the markov chain Xn. Find also P^2 and P(X2 = 6).
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Three boys A, B and C are throwing a ball to each other. A always throws the ball to B and B always throws the ball to C, but C is just as likely to throw the ball to B as to A. Show that the process is Markovian. Find the transition matrix and classify the states. Do all the states are ergodic?
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The transition probability matrix of a Markov chain is given. Is the matrix irreducible?
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