Consider the budworm population dynamics governed by the equation

where ,k the carrying capacity, and r, the birth rate of the budworm population, are positive parameters. Find out the steady states and use the perturbation to do the stability analysis of the equation for 0 < r < 1.

Consider the data showing observations on the quantity demanded of a certain commodity depending on commodity price and consumers’ income:

Find the multiple regression equation that best fits the data.

Do the stability analysis of the trivial equilibrium solution of the following competing species model:

where D1 and D2 are diffusion coefficients of the two population densities N1 and N , 2 respectively. 1 a is the growth rate, 1 b is the predation rate, 1 d is the death rate and 1 c is the conversion rate. The initial boundary conditions are

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

Formulate the model for which the reproductive function of the cancer cells in the tumor  surface is given by 2 1 ; c 1(2 c)2 3 c2 )c( ≠ − − φ = together with initial conditions 5 c = 20×10 at t = .0 Also find the density of the cancer cells in the tumour’s surface area at t = 20 days.

Return distribution of two securities are given below:

Find which security is more risky in the Markowitz sense.

A company has factories at  F₁ ,F₂ and F3 that supply products to warehouses at W_1, W₂ , and W . 3 The weekly capacities of the factories are 200, 160 and 90 units, respectively. The weekly warehouse requirements are 180, 120 and 150 units, respectively. The unit shipping costs (in ₹) are as follows:

Determine the optimal distribution for this company in order to minimize its total shipping cost.

Ships arrive at a port at the rate of one in every 4 hours with exponential distribution of inter arrival times. The time a ship occupies a berth for unloading has exponential distribution with an average of 10 hours. If the average delay of ships waiting for a berth is to be kept below 14 hours, how many berths should be provided at the port?

Consider the discrete time population model given by  N_t+1= for a population, where r is the intrinsic growth rated, b is a positive parameter. Determine the non-negative steady-state and discuss the linear stability of the model for 9 < r < .1 Also find the first bifurcation value of the parameters.

Find the number of quantities required for estimating the expected return and standard deviation for 250 securities in Markowitz model. How many estimates are required for the securities while using single-index Sharpe model?

Following is the data for number of years students studied a subject and score he/she received in that subject:

Fit the least square line to this data. What is the score of the student who has studied for two years according to this line?

“Indifference curves of an investor cannot intersect.” Is this statement true? Give reason for your answer.

The return distribution on the two securities X and Y are given in the table below:

Find 

The daily capacity of each of the factory is 150 per day and the daily requirement over each target store is 200. Find the allocation for each factory to each target store which minimize the total transport cost.

The transportation cost of 600 tons of a certain type of material from four factories  B_1, B₂ , B₃ , and B4 to three target stores  T_1,T₂ , and T₃ are given in the following table:

The transportation cost of 600 tons of a certain type of material from four factories  B_1, B₂ , B₃ , and B4 to three target stores  T_1,T₂ , and T₃ are given in the following table:

A locality is served by two malls. Each mall has two counters to serve the customers. Both the malls are equally popular and are known to have equal shares of the market. This is evident from the fact that customer’s arrive at each mall’s serving counter at the rate of 12 customers per hour. The average time to serve a customer is 05 minutes. Customers’ arrival is according to a Poisson distribution and the service time is exponential. To provide better service to the customers, the owners of the two malls decide to consolidate into a single larger mall. What is the effect of consolidation on the waiting time of customers?

Let ),t(P measured in kg, be the total mass or biomass of the fish population in a point at time t . Write the continuous model for the population growth using logistic equation. The intrinsic growth rate r and the carrying capacity k are given the values 70.0 per year and 6 80 7. ×10 kg respectively. If the initial biomass is P 25.0 0 = K, find the biomass after 2 years. Also find the time t , for which t(P ) 75.0 1 = K.

A particular component in a machine is replaced instantaneously on failure. The successive component lifetimes are uniformly distributed over the interval ]5,2[ years. Further, planned replacements take place every 3 years.

i) long-terms rate of replacements.
ii) long-terms rate of failures. Compute

Consider three random variables X₁ X₂ ,X₃ , having the covariance matrix

Write the factor model, if number of variables and number of factors are 3 and 1 respectively.

A box contains two coins: a regular coin and one fake two-headed coin. One coin is chosen at random and tossed twice. The following events are defined: (5)
 A: first coin toss results in a head.
 B: second coin toss results in a head.
 C: coin 1 (regular) has been selected.
 Find (P C|A ), (P |B C), (P A ∩ C|)B ), (P A), (P )B and (P A∩ ).B