For a mesokurtic distribution with standard deviation 5, find fourth central moment m₄.
b) The probability that a card will have a flat tyre while crossing a certain bridge is 0.00005. Find the probability that, among 10,000 cars crossing the bridge,
i) exactly two cars will have a flat tyre.
ii) at most two cards will have a flat tyre.

Let E₁, E₂, E₃ and E₄ be arbitrary events. Write the following events in set notations:
i) not more than one of E₁, E₂, E₃, E₄.
ii) one and only one of E₁, E₂, E₃, E₄.
iii) E₁ and at least one of E₂, E₃, E₄.
iv) none of E₂, E₃ and E₄ using E₁.
b) Let the probability density function of r.v. X be

and if  and , find Cov(u,v). Also check the independence of u and v.

The mean and standard deviation of a variable x are m and  respectively. Obtain the mean and standard deviation of , where a, b and c are constants.
b) If X is a random variable such that  and , determine a lower bound for P(-2 < X < 8)

 Consider the joint probability density function
$
Are both x and y regressions linear? Give reasons for your answer.

A die is thrown 60 times with the following results:

Test that the die is unbiased at 5% level of significance. Given that at 5, 6 and 7 d.f. the value of  are 11.070, 15.592 and 14.067 respectively.

6 observations on $(X, Y)$ yielded the following data:

    \sum X_i = 30, \sum Y_i = 180, \sum X_i Y_i = 1000,
    \sum X_i^2 = 200, \sum Y_i^2 = 5642.

    i)  Determine the correlation coefficient between X and Y.
    ii) Given X = 10, what will be the predicted value of Y?
    iii) Given Y = 15, what will be the predicted value of X?

he mean I.Q. of a large number of children of age 14 was 100 and standard deviation 16. Assuming that the distribution was normal, find
i) the percentage of children having I.Q. under 80.
ii) the limits in which the I.Q. of the middle 40% of the children will lie.
You may like to use the following values:

Solve the following LPP graphically:

aximize:

$
subject to the constraints:
$
$
$
b) Find all values of k for which the vectors:
$
are linearly independent.

Solve the following LPP graphically:

Maximize:

$
subject to the constraints:
$
$
$
b) Find all values of k for which the vectors:
$
are linearly independent.

Write the LPP formulation of the following assignment problem:

b) Solve the following game graphically:

Write the LPP formulation of the following transportation problem:

b) Solve the following assignment problem for profit maximization:

) For what value of k are the following vectors linearly independent?
$
b) Solve the following LP problem using simplex method:
Maximize 
Subject to 
$
$
$

) Using matrix – minima method, find the initial basic feasible solution of the following transportation problem:

Hence find the optimal solution.

b) Check whether the following set is convex:

$

 Using matrix – minima method, find the initial basic feasible solution of the following transportation problem:

Hence find the optimal solution.

b) Check whether the following set is convex:

A department has five employees with five jobs to be performed. The time (in hours) each employee will take to perform each job is given in the following matrix. How should the jobs be allocated, one per employee, so as to minimize the total man hours?

b) For the following pay-off matrix, transform the zero-sum game into an equivalent linear programming problem:

wo breakfast food manufacturers ABC and XYZ are competing for an increased market share. The pay-off matrix, shown in the following table, describes the increase in market share for ABC and decrease in market share of XYZ. Determine optimal strategies for both the manufacturers and the value of the game.

b) Find all the basic feasible solutions of the following system of linear equations:

$
$
$
Check if any of them is degenerate solution. Justify your answer.

 A company makes two kinds of leather belts. Belts A is high quality belt and belt B is of lower quality. The respective profits on A and B are ₹ 4 and ₹ 3 per belt. The production of each type A requires twice as much time as a belt. The production of each type of type B, and if all belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (both A

and B combined). Belt A require a fancy buckle and only 400 buckles per day are available. There are only 700 buckles a day available for belt B. What should be the daily production of each type of belt? Formulate this problem as an LP model and solve it by the graphical method.

b) Find the initial basic feasible solution of the following transportation problem using North-West Corner method.

 Reduce the following two person zero sum game to  game using principle of dominance. And hence solve the game.

Player B

b) Obtain the dual of the following primal LP problem:
Maximize 
Subject to 
$
$

. State which of the following statements are true and which are false. Give reasons for your answer with a short proof or a counter example.
a) The intersection of finite number of convex sets is not convex.
b) If value of the  matrix game  is 4, then .
c) If 10 is added to each of the entries of the cost matrix of a  assignment problem, then the total cost of an optimal assignment for the changed cost matrix will increase by 10.
d) For maximization LP model, the simplex method is terminated when all values .
e) The dummy source or destination in a transportation problem is added to prevent solution from becoming degenerate.

 For a given set of securities, all their portfolios lie on or within the boundary of the region shown in Fig.1.

In the feasible region, find a portfolio which has maximum return. Also, find a portfolio in this region which has minimum risk.

b) Explain the method of delineating the efficient frontier of a feasible region.

c) Given all the portfolios of n securities what criterion would an investor use to select a good portfolio?