Answer all the problems. Section A to be answered in about 500 words each and Section B to be answered in about 300 words each.
Section A:
problem 1: Describe the term differential equation? How do you apply differential equations in Economics? Discuss the role of initial condition in solving a differential equation. If your objective is to observe the stability of equilibrium, show with the help of an illustration, how a second-order differential equation can address your concern.
problem 2: Describe the term Poisson distribution? Does it have a probability density function? Why or why not? Discuss your answer in the context of the mean and variance of Poisson distribution. Give illustrations of the problems where you can make use of Poisson distribution.
Section B:
problem 3: Describe the relevant considerations of making a choice between one-tailed and two-tailed tests. How would you find out the level of significance in the above tests?
problem 4: A linear programming problem is as follows:
Max z = 30x_{1} + 50 x_{2}
Subject to
x_{1} + x_{2} ≥ 9
x_{1} + 2x_{2} ≥ 12
x_{1} ≥ 0, x_{2} ≥ 0
Find its optimal solution.
problem 5: How would you find out linear dependence of a matrix? Define the rank of a matrix in terms of its linear independence.
problem 6: The correlation coefficient between nasal length and stature for a group of 20 Indian adult males was found to be 0.203. Test whether there is any correlation between the characteristics in the population.
problem 7: prepare brief notes on the given:
a) Eigenvectors and Eigen values.
b) Taylor’s expansion.
c) Mixed strategy equilibrium.
d) Kuhn-Tucker condition.