RGU Question Paper Mathematics Semester VI 2020: Mathematical Statistics and Linear Programming

 

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B.Sc (VI Semester) Examination

MATHEMATICS

Paper : MATH–364 (New Course)

( Mathematical Statistics and Linear programming )

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Note : 1. Answer all questions.
2. The figures in the margin indicate full marks for the questions.

1. Answer any five parts : 2×5=10

(a) Write the Bayes’ theorem which enables us to find the probability/probabilities of the various events $A_1, A_2, ..., A_n$ that can cause A to occur.

(b) What do you mean by random variables? Describe discrete and continuous variables.

(c) Write the axiomatic and empirical definitions of probability.

(d) Describe briefly the range, quartile, decile and percentile of a discrete data.

(e) Define correlation. Write the formula for correlation coefficient with its limitations.

(f) Define convex set with example. Write the properties of convex sets.

(g) Define the optimal basic feasible solution of a linear programming
problem.

2. Answer any four parts : 5×4=20

(a) Calculate the mean and the standard deviation for the following
distribution :

(b) A box contains 3 blue and 2 red marbles while another box contains 2 blue and 5 red marbles. A marble drawn at random from one of the boxes turns out to be blue. What is the probability that it came from the first box?

(c) Define variance. Find the variance and the standard deviation of the sum obtained in tossing a pair of fair dice.

(d) Define moments. A random variable X has density function given by

Find the moment generating function and the first four moments about the origin.

(e) There are two factories located one at place P and the other at place Q. From these locations, a certain commodity is to be delivered to each of the parts having depots situated at A, B and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are respectively 8 and 6 units. The cost of transporting per unit is given below :

How many units should be transported from each factory to each depot in order that the transportation cost is minimum? What will be the minimum transportation cost?

(f) Fit a curve of the form $y=ax+b{x}^2$ to the following data by the method of least squares :

(g) Determine graphically the minimum value of the objective function Z = -50x + 20y subject to the constraints

3. Answer any five parts : 10×5=50

(a) The density function of a continuous
random variable X is
$f\left(x\right)=\left\{\begin{array}{l}4x(9-x^2)/81, 0\le x\le 3\\ 0, otherwise\end{array}\right.$

Find the mean, mode, median and compare them. Also find the semi-interquartile range and the mean deviation for the above density function.

(b) What do you mean by dispersion? Define skewness and kurtosis with suitable diagrams. Find the skewness and kurtosis for the following distribution defined by the normal curve, having density

(c) (i) The variance of n observations is ${\sigma }^2$. Show that if each observation is increased by k, then the variance of the new set of observations remains ${\sigma }^2$. 5

(ii) Find the mean deviation from the median for the following data : 5

(d) Describe the basic solution and basic feasible solution of an LPP. When the solution will be degenerated? Obtain all the basic solutions to the following system of llinear equations :

(e) The joint probability function of two discrete random variables X and Y is given by
$f(x,y)=c(2x+y)$
where x and y can assume all integers such that $0\le x\le 2, 0\le y\le \mathrm{3 }$and $f(x,y)=0$; otherwise.
Find—
(i) the value of the constant c;
(ii) the $P(X\ge 1, Y\ge 2)$;
(iii) P(X=2, Y=1);
(iv) Var(X) and Var(Y);
(v) Cov (X, Y).

(f) (i) What do you mean by curve fitting? What is its main purpose? Describe the method of least squares using suitable diagram. 5

(ii) Derive the formula for least squares line. Express it in terms of sample variance and covariance. 5

(g) (i) Fit a least-squares line to the following data using x as independent variable : 5

(ii) Derive the normal equations for the least-squares parabola: 5
y = a + bx + cx 2 

(h) (i) Fit a least-squares parabola having the form $y=a+bx+c{x}^{\mathrm{2 }}$for the following data : 5

(ii) The following table shows the respective heights x and y of a sample of 12 fathers and their oldest sons. Calculate the Spearman’s rank correlation coefficient : 5

(i) (i) Describe the simplex method. Find all the basic feasible solutions of the following linear programming problem without using the simplex algorithm : 7

(ii) Define cost vector associated with the basic feasible solution with suitable example. 3