Normal probability distribution function

Question

QUANTITATIVE METHODS:

Answer all the questions. Section A to be answered in about 500 words each and Section B to be answered in about 300 words each.

Section A:
 
Question 1: Consider the utility function u = f (x₁.....x_n) where x_i, i = 1,2,……., n are the quantities of the n goods consumed. Let the price of good x_i < p_i, i = 1, 2,…….., n. Let M be the consumer’s income. Show that the Lagrangian multiplier of the utility maximization problem equivalents the marginal utility of income.

Question 2:

a) What is the normal probability distribution function? Also state its properties.

b) The concentration of impurities in a semiconductor used in the production of microprocessors for computer is a normally distributed random variable with mean 127 parts per million and standard deviation 22 parts per million. The semiconductor is acceptable only if its concentration of impurities is below 150 parts per million. What is the proportion of semiconductors which are acceptable for use? (The area beneath the standard normal curve for the value of Z = 1.5 is 0.668).
 
Section B:
 
Question 3: A monopolist firm’s demand curve is given by P = 100-2q.

a) Find out its marginal revenue function.
 
b) What is the relationship between the slopes of average and marginal revenue curves?
 
c) At what price is the marginal revenue 0 (zero)?
 
Question 4: 

a) Describe the Hawkins-Simon condition for the viability of an input-output table.
 
b) Given below is the input matrix (A) and the final demand (D) vector.

Write down the specific input-output matrix for this model. Check if the given data satisfy the Hawkins-Simon-condition.

Question 5: Find the extreme value(s) of z = 2x₁² -x₁x₂ + 4x₂² + x₃ + x₃² +2?

By using the Hessian matrix check if the extreme value(s) is/are maximum or minimum.
 
Question 6: Assume that x has the given probability distribution.

Question 7: Describe the method of maximum likelihood for estimating the value of a population parameter.