In a species of animals a constant fraction of the population = 6.2 are born each breeding season and a constant fraction ẞ=4.5 die. Formulate a difference equation for the population and find out the number of individuals after fifteen seasons given that the initial number is 987. Find the closed form solution of the formulated difference equation. If the growth rate of the population is represented by r then interpret the solution obtained when i) r > 0 and ii) r < 0.
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Let P=(w1,w2) be a portfolio of two securities. If variance of P is minimum then find the value of w₁and w₂ in the following situations.
i) ρ12 = −1
ii)σ1 = σ2
iii)ρ12 = − ,5.0 σ1 = 5.1 and σ2 = .5.2
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 1 every 6 minutes and service time follows exponential distribution with a mean of 10 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.
For a multi-channel queuing system with λ = 12/hours, u=5/hours, c=3, po, = 0.056, calculate
i) The average time a customer is in the system
ii) The average number of customers in the system
iii) Whether any time would be saved for customers if the three-channel system with the service rate of 5 per hour is replaced by a single-channel system with an average service rate of 15 per hour?
In a species of animals a constant fraction of the population = 6.2 are born each breeding season and a constant fraction ẞ=4.5 die. Formulate a difference equation for the population and find out the number of individuals after fifteen seasons given that the initial number is 987. Find the closed form solution of the formulated difference equation. If the growth rate of the population is represented by r then interpret the solution obtained when i) r > 0 and ii) r < 0.
A model for insect populations leads to the difference equations
where λ and a are positive constants.
i) Write the equation in the form , and hence identify the growth rate.
ii) Plot the graph of R(Nk) as a function of N2
iii) Express the intrinsic growth rater and the carrying capacity K, for this model, in terms of the parameters, a and λ.
iv) Find the steady-state solution of this model and analyse the solution.
Consider the following data
| x | 2 | 3 | 4 | 5 | 6 |
| y | 8.3 | 16.5 | 30.2 | 65.2 | 125.6 |
Assume that the return distribution on the two securities X and Y be as given below:
| Market 1 | Market 2 | Market 3 | |
| Probability | 2.0 | 5.0 | 3.0 |
| Security X | − 20% | 18% | 40% |
| Security Y | −10% | 20% | 15% |