Download JNTUK M-Tech 2020 R19 IT Discrete Mathematical Structures Model Question Paper

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Time: 3 Hrs Max. Marks 75

(M19IT1101)

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I M.Tech I SEMESTER (R19) Regular Examinations

DISCRETE MATHEMATICAL STRUCTURES

Department of Information Technology

Answer ONE question from EACH UNIT

All questions carry equal marks

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UNIT - I

1. a) Solve for the value of c, distribution function of X and P(X = 3), given f(x) = c/x for x=1,2,3....... n as the probability function of the random variable X.
2. b) The joint probability function of two discrete random variables X and Y is given by f(x,y) = c (2x +y) where X and Y can assume all integers such that 0 < x < 2, 0 < y <= 3 and f(x,y) =0 otherwise. Solve for 1) the value of c ii) E(X) 1i1) E(Y) 1v) Var(X) and Var(Y).

(OR)

1. Let X and Y have joint density function
   F(x,y) = {2e^-(x+y) for x > 0, y > 0
   

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   0 otherwise
   Then find conditional expectation of (i) Y on X (ii) X on Y

UNIT - II

1. a) It has been claimed that in 60% of all solar installations, utility bill reduced to by one-third. Identify the probabilities for the utility bill reduce by at least one-third (1) in fr of five installations and (ii) at least fr of five installations
2. b) Utilize probability mass function of Poisson’s distribution to determine its mean, variance, coefficient skewness & Kurtosis.

(OR)

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1. a) If 20% of memory chips made in a certain plant are defective, then identify the probabilities, that a randomly chosen 100 chips for inspection (i) at most 15 will defective (ii) at least 25 will be defective (iii in between 16 and 23 will be defective
2. b) Make use of pdf of the Exponential distribution to find its mean and variance

UNIT - III

1. a) The following table shows corresponding values of three variables X, Y, Z. Model the least square regression equation Z= a+bx-+cy
   X | 1 | 2 | 1 | 2 | 3
   Y | 2 | 3 | 1 | 1 | 2
   

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   Z | 12 | 19 | 8 | 11 | 18
2. b) Explain the procedure for fitting an exponential curve of the form y = a^x

(OR)

1. a) What the properties of a good estimator. Explain each of them
2. b) Suppose that n observations X_i, X₂, X₃ are made from normal distribution and variance is unknown. Identify the maximum likelihood estimate of the mean.

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UNIT - IV

1. a) Show that in any non-directed graph there is even number of vertices of odd degree.
2. b) State and prove Euler’s formula for planar graphs

(OR)

1. a) Prove that a tree with ‘n’ vertices have ‘n-1’ edges
2. b) If T is a binary tree of n vertices, show that the number of pendant vertices is (n+1)/2

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UNIT - V

1. a) Make use of the principles of Inclusion and exclusion find the number of integers between 1 and 100 that are divisible by 2, 3 or 5
2. b) Identify the number of integral solutions for x₁ + x₂ + x₃ + x₄ + x₅ = 50 where x₁ > 4, x₂ > 7, x₃ >= 14, x₄ >= 10, x₅ > 0

(OR)

1. a) Solve the recurrence relation a_n - 7a_n-1 + 12a_n-2 = 0 for n>2 using Generating function method.
2. b) Solve a_n — 7a_n-1 + 10a_n-2 = 4ⁿ for n>2.

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