JNTU
JNTU B.Tech III PROBABILITY AND STATISTICS Previous Question Paper (May - 2018 - 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
What is a random variable? Give an example.
Answer
A random variable is a real-valued function that assigns a numerical value to each outcome in the sample space of a random experiment. It maps the set of possible outcomes to the set of real numbers, allowing for the application of mathematical and statistical methods to probabilistic events.
For example, in the experiment of tossing a fair coin twice, if we define the random variable X as the number of heads obtained, the possible values for X are 0, 1, and 2, corresponding to the outcomes {TT}, {HT, TH}, and {HH} respectively.
Find the moment generating function of Binomial distribution.
Answer
For a binomial random variable X with parameters n and p, the probability mass function is given by P(X = k) = nCk pk qn-k, where q = 1 - p. The moment generating function MX(t) is defined as the expected value of etX.
By grouping the terms with the same exponent k, we obtain:
Applying the Binomial Theorem, which states that (a + b)n = Σk=0n nCk ak bn-k, we identify a = pet and b = q:
The joint density function of X and Y is f(x, y) = a if 1 ≤ x ≤ 3, 2 ≤ y ≤ 4, otherwise 0. Find 'a'.
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The two regression lines are 7x - 16y + 9 = 0 and 5y - 4x - 3 = 0. Find the mean values x and y.
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State central limit theorem.
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For F-distribution, find F0.99(28, 12).
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Define reneging.
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Write the formula for the probability distribution density function of the waiting time distributions.
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Define discrete random process.
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What is Markov chain?
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Part B
A random variable X has the following distribution. X: 0, 1, 2, 3, 4. P(X): C, 2C, 2C, C, 5C2. Find (a) C (b) P(1 < X < 4) (c) P(X < 3) (d) mean and (e) variance of X.
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Compute the first four moments about the mean from the following data: x: 1, 2, 3, 4, 5; f: 2, 3, 5, 4, 1
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If X is normally distributed with mean 2 and variance 0.1, find P(|X-2| ≥ 0.01).
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The joint density function of the two random variables X and Y is given by f(x, y) = (x+y)/3, 0 ≤ x ≤ 1, 0 ≤ y ≤ 2; otherwise 0. a) Are the random variables X and Y independent? b) Find conditional density functions. c) Check whether the conditional density functions are valid or not.
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Two independent variables X and Y have means 5 and 10 and variances 4 and 9 respectively. If U = 3x+4y, V = 3x-y, find the coefficient of correlation between U and V.
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Calculate the rank correlation coefficient from the following data: X: 1, 2, 3, 4, 5, 6, 7; Y: 2, 4, 5, 3, 8, 6, 7.
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A random sample of size 100 is taken from a population with standard deviation σ = 51 and sample mean x = 21.6. Construct 95% confidence interval for the population mean μ.
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Explain briefly (i) type - I error and (ii) type-II error.
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Random samples of 400 men and 600 women were asked whether they would like to have a flyover near their residence. 200 men and 325 women were in favour of the proposal. Test the hypothesis that the proportions of men and women in favour of the proposal are same at 5% level of significance.
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Explain briefly the main characteristics of a queuing system.
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A car park contains 5 cars. The arrival of cars is Poisson with a mean rate of 10 per hour. The length of time each car spends in the car park has negative exponential distribution with mean 2 hours. How many cars are in the car park on average? And what is the probability of a newly arriving customer finding the car park full and having to park his car elsewhere?
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A stochastic process is described by X(t) = Asin(ωt) + Bcos(ωt), where A and B are independent random variables with zero means and equal standard deviations. Show that the process stationary of the second order.
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Show that the matrix A = [[0, 1], [1/2, 1/2]] is a stochastic matrix. Check whether it is regular.
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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. If C was the first person to throw the ball, find the probabilities that a) The ball is with A b) The ball is with B and c) The ball is with C after three throws.
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