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?

For the (M/M/K) : (∞ / FIF )0 queuing model with arrival rate λ and service rate per service channel µ, obtain the steady-state probability of n customers in the system, P . n Hence or otherwise, find the probability that an arrival has to wait.

Consider the following system considering of two switches I and II between two points A and B:

A signal is sent from the point A to point B and is received at B if both the switches I and II are closed. It is assumed that the probabilities of I and II being closed are 0.8 and 0.6 respectively and that P [II is closed] I is closed] = P [II is closed].

Find:

i) The probability that signal is received at B.

ii) The conditional probability that switch I was open, given that the signal was not received at B,

iii) The conditional probability that switch II was open, given that the signal was not received at B.

In a certain state, 58 landfills are classified according to their concentration of three hazardous chemicals: arsenic, barium and mercury. Suppose that the concentration of each one of the three chemicals is characterized as either high or low. If a landfill is chosen at random from among 58 landfills, given the following configuration:

Find the probability that it has:

i) high concentration of Barium,

ii) high concentration of mercury and low concentration of both Arsenic and Barium and

iii) high concentratin of any one of the chemicals and low concentration of the other two.

A study about a population showed that the mobility of population of a state to a village, town and city is in the following percentages:

What will be the proportion of population village, town and city after one year and two years, given that the present proportion of the population in the village, town and city are respectively 0.50, 0.40 and 0.10?

What is sampling? Explain the different methods of probability sampling.

List the different operational process that need to be done to bring data into a presentable form and explain each process with suitable examples.

Describe the various steps adopted in test construction in order to arrive at accuracy in the result.

Explain the different components of a project proposal.

Describe the concept of community ownership in CSR activities with suitable examples.

Analyse the role of CSR in achieving women's empowerment in India.

Discuss the different sections under Companies Act of 2013 related to CSR?

Write Essay on the perspective of CSR in global context, with special reference to developed countries.

Explain any two key international conventions and financing mechanisms for the conservation and sustainable management of aquatic ecosystems, with special reference to developing countries.

Identify the measures for research and capacity building initiated by government for achieving sustainable use of freshwater and marine aquatic ecosystems in India.