RGU Question Paper Mathematics Semester VI 2021: Mathematical Statistics and Linear Programming

 June-2021

B.A/B. Sc. (VI Semester) Examination

MATHEMATICS

Paper: MATH-364 (N/C)

(Mathematical Statistics and Linear Programming)

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Notes: (i) Answer all questions.

(ii) The figures in the margin indicate full marks for the questions.

I. Answer any five parts: 2x5=10

(a) Write short note on Skewness and Kurtosis.

(b) Why we study the notion – measure of central tendency, like mean, median and mode?

(c) Write short note on slack and surplus variables.

(d) Define probability axiomatically. Explain it a suitable example.

(e) Give the physical interpretation of correlation in your own language.

(f) Define convex set with suitable examples.

(g) State the fundamental theorem of L.P.P.

II. Answer any four parts: 5x4=20

(a) Find the median for the following distribution:

(b) Solve, graphically, the following L. P. P.:

Maximize $z=x_1+2x_2$ subject to the constraints:

$x_1+x_2\le 3$, $x_1+2x_2\le 5$, $3x_1+x_2\le 6$ and $x_1,x_2\ge 0$. Is the solution is unique?

(c) Define convex hull. Prove that the set of all convex combinations of a finite number of points of $S\subset {\mathbb{R}}^n$ is a convex set.

(d) Find the probability of not getting a 7 or 11 total on either of two tosses of a pair of fair dice.

(e) Prove that for an impossible event $\varphi$, the probability $P\left(\varphi \right)=0$. Also prove that if A and B are any two events, then

$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$.

(f) Find all the basic feasible solutions of the following equations:

$x_1+7x_2+2x_3+x_4=3, 7x_1+x_2+3x_3+2x_4=10$.

III. Answer any five parts: 10x5=50

(a) Use the Simplex method to solve the L. P. P.:

Maximize $z=2x_1+4x_2+x_3+x_4$

subject to the constraints:

$x_1+3x_2+x_4\le 4, 2x_1+x_2\le 3, x_2+4x_3+x_4\le 3, x_1,x_2,x_3,x_4\ge 0$.

(b) Define extreme points. What is the relation between basic feasible solutions and extreme points? Explain the different steps in formulating a linear programming problem. What are the different forms of LPPs? Explain each of them with suitable examples.

(c) What do you mean by mode? Compute the mode of the following distribution:

Give a brief description of quartile, decile and percentile with suitable examples.

(d) What do you mean by measure of dispersion? Explain it with suitable examples. Calculate the standard deviation for the following table:

(e) What do mean by curve fitting? Write differences between the linear and nonlinear curve fitting. What is regression? Explain the method of least squares with suitable example.

(f) Write a short note on correlation and correlation coefficient. Find the correlation coefficient between the variables x and y of the following data:

(g) Following table shows the respective height x and y of a sample of 12 fathers and their oldest sons.

(1) Construct a scatter diagram.

(2) Find the least-squares regression line of y on x.

(3) Find the least-squares regression line of x on y.

(h) Write a short note on conditional probability. Briefly describe the Bayes’ theorem with suitable example. Two cards are drawn from a well-shuffled ordinary deck of 52 cards. Find the probability that they are both aces if the first card is (1) replaced, (2) not replaced.