Solve your IGNOU Doubts

Question:

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.

See Answer →

Question:

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?

See Answer →

Question:

Maximize $Z = 5x_1 + x_2 + 3x_3$ , subject to the constraints
$-x_1 + 2x_2 + x_3 \le 4, 4x_2 - 3x_3 \le 2, x_1 - 3x_2 + 2x_3 \le 3$ and $(x_1, x_2, x_3) \ge 0$ and are integers.

See Answer →

Question:

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

$$\frac{\partial N_1}{\partial t} = a_1 N_1 - b_1 N_1 N_2 + D_1 \frac{\partial^2 N_1}{\partial x^2} - v_1 \frac{\partial N_1}{\partial x}$

$$\frac{\partial N_2}{\partial t} = -d_1 N_2 + C_1 N_1 N_2 + D_2 \frac{\partial^2 N_2}{\partial x^2} - v_2 \frac{\partial N_2}{\partial x}, \quad 0 \le x \le L$

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:

$N_i(x, 0) = f_i(x) > 0, \quad 0 \le x \le L, \quad i = 1, 2.$

$N_i = \overline{N}_i$ at $x = 0$ and $x = L \forall t, \quad i = 1, 2.$

where $\overline{N}_i$ are the equilibrium solutions of the given system of equations.

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

See Answer →

Question:

Do the stability analysis of the following model formulated to study the effect of toxicant on prey-predator population and interpret the solution.
$$\frac{dN_1}{dt} = r_0 N_1 - r_1 C_0 N_1 - b N_1 N_2$
$$\frac{dN_2}{dt} = -d_0 N_2 - d_1 V_0 N_2 + \beta_0 b N_1 N_2$
$$\frac{dC_0}{dt} = k_1 P - g_1 C_0 - m_1 C_0$
$$\frac{dV_0}{dt} = k_2 P - g_2 V_0 - m_2 V_0$
$$\frac{dP}{dt} = Q - hP - kP(N_1 + N_2) + gC_0N_1 + \ell V_0 N_2 \text{.}$

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

$Q = \text{constant input rate}$

$h = \text{decay rate}$

$P(t) = \text{environmental toxicant concentration}$

$k = \text{ingestion rate of toxicant by the populations}$

$g, \ell = \text{return rate of toxicant in the environment after the death of the populations,}$
$\text{assuming that toxicant is non-degradable}$

$Q > 0, h, k, g, \ell \text{ are positive constants.}$

See Answer →

Question:

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

$$x_{n+1} = x_n \exp \left[ r \left( 1 - \frac{x_n}{K} \right) \right],$

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 $r = 2 + \varepsilon$ , where $0 < \varepsilon \ll 1$ . Since a species becomes extinct if $x_n \leq 1$ for any n > 1, show using iterations, that irrespective of the size of r > 1 the species could become extinct if the carrying capacity $k < r \exp[1 + e^{r-1} - 2r]$ .

See Answer →

Question:

Consider a discrete model given by

$$N_{t+1} = \frac{r N_t}{1 + b N_{t-1}^2} = f(N_t), r > 1.$
Investigate the linear stability about the positive steady state N by setting $N_t = N^</em> + n_t$ . Show that n_t satisfies the equation
$$n_{t+1} - n_t + 2(r - 1)r^{-1}n_{t-1} = 0.$

Hence show that $r = 2$ is a bifurcation value and that as $r \rightarrow 2$ the steady state bifurcates to a periodic solution of period 6.

See Answer →

Question:

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

Husband’s age (years)	Wife’s age (years)	Husband’s age (years)	Wife’s age (years)
32	27	51	50
25	30	48	46
36	34	37	36
72	65	50	42
37	37	51	46
36	38	36	35

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.

See Answer →

Question:

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:

Customer Orders (in thousands)	Add-delete items (in thousands)	CPU time (in hours)
123.5	2.108	141.5
146.1	9.213	168.9
133.9	1.905	154.8
128.5	0.815	146.5
151.5	1.061	172.8
136.2	8.603	160.1
92.0	1.125	108.5

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

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

See Answer →

Question:

Let $\text{P} = (w_1, w_2)$ be a portfolio of two securities X and Y. Find the values of w₁ and w₂ in the following situations:

i) $\rho_{xy} = -1$ and P is risk free.

ii) $\sigma_x = \sigma_y$ and variance P is minimum.

iii) Variance P is minimum and $\rho_{xy} = -0.6, \sigma_x = 3$ and $\sigma_y = 5$ .

See Answer →

Question:

Return distributions of the two securities are given below:

Return		Probabilities
X	Y	`P_xj=P_yj=P_j`
0.20	0.15	0.30
0.15	0.08	0.25
0.10	0.05	0.15
0.11	0.09	0.25

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

See Answer →

Question:

A tumour is developing from the organ of a human body with concentration $5.2 \times 10^9$ with growth and decay control parameters 7.2 and 2.7 respectively. In how many days the size of the tumor will be twice?

See Answer →

Question:

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 $g / \text{min}$ . 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 $t = 0, G = 70\text{g}$ then

i) formulate the model.

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

iii) discuss the long term behavior of the model.

See Answer →

Question:

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

Portfolio	Expected return	Standard deviation
W	10%	25%
X	5%	7%
Y	17%	37%
Z	12%	13%

See Answer →

Question:

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.

See Answer →

Question:

Consider the two parents which are participating in partially mapped cross over as shown below:

Parent 1: C D | E A B |I H G F

Parent 2: A B | C D E |F G H I

Using partially mapped crossover assuming 2^nd and 6^th as the cross over sites, find the children solution.

See Answer →

Question:

Computer the output for the neurons in the kohonen networks, the related data is given below:
i) Input to Kohnen neural network:
Input Neuron-1 $(I_1) = 0.5$
Input Neuron-2 $(I_2) = 0.75$
ii) Connected weights between the neurons are as given below:
$I_1 \rightarrow O_1 : 0.1$
$I_2 \rightarrow O_1 : 0.2$
$I_1 \rightarrow O_2 : 0.3$
$I_2 \rightarrow O_2 : 0.4$

See Answer →

Question:

Using diagram, show the difference between feed-forward neural network and recurrent neural network.

See Answer →

Question:

Consider the following table for the connections between input neurons and the hidden layer neurons:

Input Neurons	Hidden Layer Neurons	Connection Weight
1	1	- 1
1	2	- 0.1
1	3	1
2	1	- 1
2	2	1
2	3	1
3	1	- 0.2
3	2	- 0.3
3	3	- 0.6

The connection weights from the hidden layer neurons to the output neurons are -0.6, -0.3 and -0.6, for the first, second and third neurons, respectively.

Corresponding threshold value for the output layer is 0.5 and for the hidden layer is 1.8, 0.005 and 0.2 for the first, second and third neurons, respectively.
i) Draw the diagram of the network.

ii) Write the output at each node.

See Answer →

Question:

Determine the $\alpha$ -cut of the fuzzy set (A) are given below, at 0.7 and 0.2.
$$A = \left\{ \frac{0}{10}, \frac{0}{20}, \frac{0.2}{30}, \frac{0.8}{40}, \frac{1.0}{50}, \frac{1.0}{60}, \frac{0.6}{70}, \frac{0.2}{80}, \frac{0}{.90}, \frac{0}{100} \right\}$
Also, compare the $\alpha$ -cut of the two outcomes, and give comments for status of $\alpha$ -value variation.