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.

What is the purpose of principal component analysis? Given the covariance matrix of order 2× ,2 explain how would you extract the principal components. Also, explain how would you find the proportion of total population variance for all the principal components.

In an investigation related to a specific type of scores of men and women aged 65 to 70, the mean verbal and performance scores for 101 subjects were found to be:

The sample covariance matrix of the scores was

In order to test the null hypothesis that observations came from a population with mean vector   apply a suitable test statistic. You may consider α = 0.01 for the test.

It is claimed that the function     is the joint 16

distribution function of the random variables X and Y. Then:

i) determine the corresponding joint probability density function f_x,y and

ii) calculate the probability  

Let {X_n; }0 n ≥ be a Markov chain with four states; 1, 2, 3, 4 and the following transition probability matrix:

i) Find the probability P[X₃, =3,X₂ =1|X₁ = 2].

ii) Classify the states of the given Markov chain.

The joint distribution of the random variables X and Y is given by:

where α,β > 0 with α + β = 4/1 .

i) Derive the marginal distribution of X and Y.

ii) Calculate the E(X), E(Y) and E(XY).

iii) Show that Cov(X,Y) = 0.

iv) Show that the variables X and Y are dependent.

The certain item is manufactured by three factories, say 1, 2 and 3. It is known that 1 turns out twice as many items as 2 and that 2 and 3 turn out the same number of items (during a specified production period). It is also known that 2 percent of the items produced by 1 and 2 are defective while 4 percent of those manufactured by 3 are defective. All the items produced are put into one stock pile and then one item is chosen at random. The chosen item was found defective. What is the probability that it was produced in factory 1?