Let the joint probability density function of two discrete random X and Y be given as:

ii) Find the marginal distribution of X and Y.

iii) Find the conditional distribution of X given .

iv) Test the independence of variable s X and Y.

v) Find .

 Let (X, Y) have the joint p.d.f. given by:

i) Find the marginal p.d.f.'s of X and Y.

ii) Test the independence of X and Y.

iii) Find the conditional distribution of X given .

iv) Compute  and .

Let the random vector  has mean vector [-2, 3, 4] and variance
covariance matrix . Fit the equation . Also obtain the multiple correlation coefficient between X₃ and [X₁, X₂].
b) Define ultimate extinction in a branching process. Let 
0 < b < c < b + c < 1 and . Then discuss the probability of extinction in different cases for  or E(X₁) < 1.

 At a certain filling station, customers arrive in a Poisson process with an average time of 12 per hour. The time interval between service follows exponential distribution and as such the mean time taken to service to a unit is 2 minutes. Evaluate:
i) Probability that there is no customer at the counter.

ii) Probability that there are more than two customers at the counter.
iii) Average number of customers in a queue waiting for service.
iv) Expected waiting time of a customer in the system.
i) Probability that a customer wait for 0.11 minutes in a queue.

 Consider a Markov chain with transition probability matrix:

i) Whether the chain is irreducible? If irreducible classify the states of a Markov chain i.e., recurrent, transient, periodic and mean recurrence time.

ii) Find the limiting probability vector.

State whether the following statements are True or False. Justify your answer with a short proof or a counter example: 
a) If P is a transition matrix of a Markov Chain, then all the rows of  are identical.
b) In a variance-covariance matrix all elements are always positive.
c) If X₁, X₂, X₃ are iid from , then  follows .
d) The partial correlation coefficients and multiple correlation coefficients lie between -1 and 1.
e) For a renewal function .

 A library wants to improve its service facilities in terms of the waiting time of its borrowers. The library has two counters at present and borrowers arrive according to Poisson distribution with arrival rate 2 every 10 minutes and service time follows exponential distribution with a mean of 15 minutes. The library has relaxed its membership rules and a substantial increase in the number of borrowers is expected. Find the number of additional counters to be provided if the arrival rate is expected to be twice the present value and the average waiting time of the borrower must be limited to half the present value. 

Ships arrive at a port at the rate of one in every 6 hours with exponential distribution of inter-arrival times. The time a ship occupies a berth for unloading has exponential distribution with an average of 12 hours. If the average delay of ships waiting for berths is to be kept below 15 hours, how many berths should be provided at the port? 

 Maximize  , subject to the constraints
 and  and are integers. 

 Do the stability analysis of the following competing species system of equations with diffusion and advection

where V₁ and V₂ are advection velocities in x direction of the two populations with densities N₁ and N₂ respectively. a₁ is the growth rate, b₁ is the predation rate, d₁ is the death rate, C₁ is the conversion rate. D₁ and D₂ are diffusion coefficients. The initial and boundary conditions are:

 at  and 

where  are the equilibrium solutions of the given system of equations.

Interpret the solution obtained and also write the limitations of the model. 

Do the stability analysis of the following model formulated to study the effect of toxicant on prey-predator population and interpret the solution.

Where all the variables and constants are same as defined in the system (32)-(35) except for the following

 The population dynamics of a species is governed by the discrete model

where r and k are positive constants. Determine the steady states and discuss the stability of the model. Find the value of r at which first bifurcation occurs. Describe qualitatively the behaviours of the population for , where . Since a species becomes extinct if  for any n > 1, show using iterations, that irrespective of the size of r > 1 the species could become extinct if the carrying capacity .

Consider a discrete model given by

Investigate the linear stability about the positive steady state N by setting . Show that n_t satisfies the equation

Hence show that  is a bifurcation value and that as  the steady state bifurcates to a periodic solution of period 6.

 The population consisting of all married couples is collected. The data showing the age of 12 married couples is as follows:

i) Draw a scatter plot of the data 

ii) Write two important characteristics of the data that emerge from the scatter plot. 

iii) Fit a linear regression model to the data and interpret the result in terms of the comparative change in the age of husband and wife. 

iv) Calculate the standard error of regression and the coefficient of determination for the data. 

 Companies considering the purchase of a computer must first assess their future needs in order to determine the proper equipment. A computer scientist collected data from seven similar company sites so that computer hardware requirements for inventory management could be developed. The data collected is as follows:

i) Find a linear regression equation that best fit the data. 

ii) Estimate the error variance for the regression model obtained in i) above. 

 Let  be a portfolio of two securities X and Y. Find the values of w₁ and w₂ in the following situations:

i)  and P is risk free.

ii)  and variance P is minimum.

iii) Variance P is minimum and  and . 

Return distributions of the two securities are given below:

Find which security is more risky in the Markowitz sense. Also find the correlation coefficient of securities X and Y.

A tumour is developing from the organ of a human body with concentration  with growth and decay control parameters 7.2 and 2.7 respectively. In how many days the size of the tumor will be twice? 

 Let G(t) be the amount of the glucose in the bloodstream of a patient at time t. Assume that the glucose is infused into the bloodstream at a constant rate of  . At the same time, the glucose is converted and removed from the bloodstream at a rate proportional to the amount of the glucose present. If at  then

i) formulate the model.

ii) find g(t) at any time t.

iii) discuss the long term behavior of the model. 

Which one of the following portfolios cannot lie on the efficient frontier as described by Markowitz?