IGNOU MMT 9 SOLVED ASSIGNMENT

Ques 1.

A company manufacturing soft drinks is thinking of expanding its plant capacity so as to meet future demand. The monthly sale for the past 6 years are available. State, giving reasons, the type of modelling you will use to obtain good estimates for future demand so as to help the company make the right decisions. Also state four essentials and four non-essentials for the problem.

Ques 2.

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

Ques 3.

 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. 

Ques 4.

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? 

Ques 5.

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.

Ques 6.

 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 . 

Ques 7.

 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. 

Ques 8.

 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. 

Ques 9.

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.

Ques 10.

 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 .

Ques 11.

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

Ques 12.

 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. 

Ques 13.

 Maximize  , subject to the constraints
 and  and are integers. 

Ques 14.

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? 

Ques 15.

 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. 

Ques 16.

A company manufacturing soft drinks is thinking of expanding its plant capacity so as to meet future demand. The monthly sale for the past 6 years are available. State, giving reasons, the type of modelling you will use to obtain good estimates for future demand so as to help the company make the right decisions. Also state four essentials and four non-essentials for the problem.

Ques 17.

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

Ques 18.

 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. 

Ques 19.

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? 

Ques 20.

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.

Ques 21.

 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 . 

Ques 22.

 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. 

Ques 23.

 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. 

Ques 24.

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.

Ques 25.

 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 .

Ques 26.

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

Ques 27.

 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. 

Ques 28.

 Maximize  , subject to the constraints
 and  and are integers. 

Ques 29.

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? 

Ques 30.

 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. 

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