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 )s(P and P )s( n respectively is the probability generating function (pgf ) of the i.i.d.

random variables and the random variables

Then show that:and

In order to fit the regression line 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 C₁ 4  = and C₂ = 6units

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

apply a suitable test statistic. You may consider α = 01.0 for the test.

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.

i) Find the probabilit 

ii) Classify the states of the given Markov chain.

b) It is claimed that the functionis the joint  distribution function of the random variables X and Y. Then:

i) determine the corresponding joint probability density functiond F_X Y,  and

ii) calculate the probability 

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?

Discuss the process of implementing marketing mix in library services.

Explain the criteria for evaluating a dictionary.

Discuss the emerging trends in database services.

Differentiate between responsive and anticipatory services.

Describe the different types of information needs.

Discuss the role of international agencies as sources of information.

Discuss in detail how will you conduct a user study?

Explain, how peripheral information professionals can perform the functions of information disseminators.