Find an optimal parenthesisation of the matrix chain product whose sequence of dimensions is 10, 25, 10, 5, 17.

Construct the Huffman code tree for the set of frequencies in the table below: 

Let me NO {0} and n∈ N. Let the god of x, y ∈ Z be denoted by (x, y). Prove that (m, n) = (m (mod n), n).

Find an optimal parenthesisation of a matrix chain whose sequence of dimensions is (5, 10, 3, 12, 15).

Determine the LCS of (1, 1, 1, 1, 1, 0, 0, 1, 1, 1) and (1, 0, 1, 0, 0, 1, 1, 0, 1, 0) using dynamic programming approach, showing all the steps.

Show the results of inserting the keys below in order into an empty B-tree of minimum degree 2:

2, 5, 4, 3, 6, 9, 8, 7, 1, 2

Construct a Binary Search Tree for the following sequence of numbers: 15, 50, 70, 60, 20, 90, 10, 40, 100, 35 with 40 as the value of the key for the root node.

Sort the following numbers using the QuickSort algorithm: 2, 9, 7, 6, 4, 3, 8, 5

Explain the Longest Common Subsequence problem with an example.

Explain the string matching problem with an example.

Define and explain the Big-O, Big - 2 and Big - notations with examples.

Discuss the different zones of centrism ideologies.

What are the different phases of socio-technical transition. Explain each phase with suitable examples.

Explain how globalization influences the economic growth of different nations across the world, with special reference to developing countries.

Discuss the limitations and future scope of the Global Biodiversity Fund and the Global Carbon Fund.

State whether the following statements are true or false. Justify your answer with a short proof or a counter example:

i) Although in one-step transition probability matrix, P, of a Markov chain the sum of each row must necessarily be unity but in higher order transition probability matrices, P^(j); j = 2,3,4,...... This rule is not necessary.

ii) For the joint pdf of random variables (X,Y) given by:

iii) Let the r.v. X follows the negative exponential distribution with parameter . Then

iv) One of the examples of non-Poisson queuing system is the (M/G/1): (∞(F1F0) queuing model.

v) If X₁,X₂,..., X, be a random sample from N_p, (μ, Σ), then maximum likelihood estimators of µ and ∑are:

A community has two police cars, which operate independently of one another. The probability that a specific car will be available when needed is 0.99.

i) What is the probability that neither car is available when needed?

ii) What is the probability that a car is available when needed?

What is branching process? Give two real examples of branching process.

If P(s) and P_n(s) respectively is the probability generating function (pgf) of the i.i.d. random variables and the random variables {X_n}, where X_n+1 = 

Then show that:

P_n(s) = P_n-1(P(s))

and P_n(s) = P_n-1(P(s)).

In order to fit the regression line y = β₁ + β₂ on a data set consisting of 34 pairs of values (z,y), the least square estimates of β1 and β2 are to be computed. From the data set, the following values are obtained:

Obtain the fitted regression line.

Distinguish between ‘Age Replacement’ and ‘Block Replacement’ policies. Let the lifetimes Y₁ ,Y₂ ,...., are independently and identically distributed random variables and follows negative exponential distribution with parameter 5. If lifetimes T > 0 and age replacement policy is to be employed, then:

i) find the mean renewal time and

ii) find the long-run average cost per unit time, given the costs C 4 1 = and C 6 2 = units of money.