Department of Mathematics

WORK-BOOKof

Mathematics- Numerical Methods and Probability Theory

(BSC011)

ForB. Tech. 2nd year- 3rd Semester

(Civil & Electrical Engineering)

Name of the Student:

Enrollment Number:

University of Engineering & Management, Jaipur

Course Code: BSC011L-T-P: 3-1-0Credits: 4

Introduction:

The aim is to teach the student various topics in Numerical Analysis such as solutions of

nonlinear equations in one variable, interpolation and approximation, numerical differentiation and

integration, direct methods for solving linear systems, numerical solution of ordinary differential

equations. The objective of Probability theory is to familiarize the students with statistical techniques

and tools at an intermediate to advanced level that will serve them well towards tackling various

problems in the discipline.

Course Outcomes:

The students will learn:

Knowledge and Understanding: Students are able to understand the nature and operations of

Numerical Analysis.

Intellectual Skills: By the end of the course the student is expected to solve real-life and

Engineering applications.

The ideas of probability and random variables and various discrete and continuous probability

distributions and their properties.

The basic ideas of statistics including measures of central tendency, correlation and regression.

The statistical methods of studying data samples.

Course contains:

Numerical Methods

Module 1: (6L)

Accuracy and Precision: Error Analysis. Solution of polynomial and transcendental equations –

Bisection method, Newton- Raphson method and Regula-Falsi method. Interpolation using

Newton’s forward and backward difference formulae. Interpolation with unequal intervals:

Newton’s divided difference and Lagrange’s formulae.

Module 2: Numerical Differentiation, Numerical integration (6L)

: Trapezoidal rule and Simpson’s 1/3rd and 3/8 rules. Ordinary differential equations: Taylor’s

series, Euler and modified Euler’s methods. Runge- Kutta method, Least square approximation.

Probability and Statistics

Module 3: Basic Probability (4L)

Definition of Probability, sampling theory, Conditional probability. Baye’s theorem.

Module 4: Probability Distribution (10L)

Random variable: Continuous and discrete random variables. Mean, Median, mode, Expectation,

Variance and standard deviation. Distributions: Bernoulli, Binomial, Poisson, Exponential &

Normal distributions.

Module 5: Statistics theory (7L)

Descriptive Statistics: Mean, Median, mode, Standard Deviation; Hypothesis testing.

Reference Books

Das N.G.: Statistical Methods, TMH.

Dutta& Jana: Introductory Numerical Analysis(All course).

Dr.B.S.Grewal:Numerical Methods in Engineering &science

Jain, Iyengar ,& Jain: Numerical Methods.

Baburam: Numerical Methods, Pearson Education.

Probability and Statistics (Schaum's Outline Series)

Module 1:

1. Using Newton’s Forward Interpolation, the area of a circle of diameter .

80 85 90 95 100

5026 5674 6362 7088 7854Calculate the area of a circle of diameter 105.

2. pply Newton’s backward Interpolation to the data below, to obtain a polynomial of degree4 in

1 2 3 4 5

1 -1 1 -1 1

3. Using Newton’s divided difference method to find from the following available data:

0 1 2 4 5 6

1 14 15 5 6 19.

4. Find as a polynomial in for the following data by Newton’s divided difference method:

-4 -1 0 2 5

1245 33 5 9 1335

5. Estimate the value of and from the following available data: 20 25 30 35 40 45

354 332 291 260 231 204Using Newton’s Forward Interpolation

6. Population of a town is given bellow

Year 1921 1931 1941 1951 1961 1971Population(in lakh)

20 24 29 36 46 51

Compute the population during the period 1955 to1961 and 1925.

7. Use Lagrange’s interpolation formula, to find the value of f(40): x: 30 35 45 55

f(x): 148 96 68 34

8. Find the missing term of the following data X 2 4 6 8 10Y 5.6 8.6 13.9 - 35.6

9.Prove a)

b)

c)

10. Evaluate

(i) (ii) (iii)

11. What is the difference between direct and iterative method and explain rate of convergence.

12. Perform first two steps of Bi-section method to find the roots of

13. Perform first step of Regula-Falsi method to find the roots of

14. For the following algebraic equation perform four iterations of Newton Rapshon’s method

.

Module 2:

1. Evaluate correct to three decimal places using trapezoidal rule of integration.

2. Compute the value of the following integral by Simpson’s 1/3 formula:

3. Compute the value of log e2

from the following integral by Simpson’s 3/8th rule and compare it with the

value obtained by actual integration. Also calculate the value of percentage error

4. Find, from the following table the area bounded by the curve and the x-axis from to ,

5. Find an approximate value(by Euler’s) of

6. Using fourth order Runge-Kutta method find the numerical solution of

at in two steps.

7. Given with initial condition at . Find for by using Taylor’s series method. (divide into four step).

8. Use Euler’s Modified method with step size .2 to find the value of

at for the following

differential equation:

9. Fit a straight line to the following data X 1 2 3 4 6 8Y 2.4 3 3.6 4 5 6

Module 3:

1. A bag contains 5 whites and 4 black balls. If 3 balls are drawn at random. What are the probabilities of:

(A) 2 of them are white (B) almost one of them is white (C) at least 2 are white.

2. The manufacture process of an article consists of two parts x and y. the probabilities of defect in part x

and y are 10% and 15% respectively. What is the probability that the assembled product will not have any

defect ?

3. A pair of dice is thrown. Find the probability of getting a sum of 7, when it is known that the digit in the

1st dice is greater than that of the second.

4. A can hit a target 4 times in 5 shots; B 3 times in 4 shots; C twice in 3 shots. They fire a target. What’s

the probability that at least two shots hits ?

5. Two urns contain respectively 5 white, 7 black and 4 white, 2 black balls. One of the urns is selected by

the toss of a fair coin and then 2 balls are drawn without replacement from the selected urn. If both balls

drawn are white, what is the probability that the first urn selected ?

6. In a bolt factory, machines A, B and C manufacture 25%, 35% and 40% respectively of the total bolts

5%, 4% and 2% are defective bolts. A bolt is drawn at random from the product and is found to be

defective. What is the probability that it was manufactured by machine A, B and C ?

7. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?

8. Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?

9. Define Exhaustive, Mutually Exclusive and Independent events.

10. A and B take turns in throwing of two dice, the first to throw 9 will be awarded a prize. If A has the first turn, show that their chances of winning are in the ratio 9:8.

11. Three groups of children contain respectively 3 girls 1 boy, 2 girls 2 boys, 1 girl 3 boys. One child is selected at random from each group. Find the probability of selecting 1 girl and 2 boys.

12. If , then prove that (i) (ii) .

13. If A and B are independent events, then prove that are also independent.

Module 4:

1. A manufacturer supplies quarter horsepower motors in lots of 25. A buyer, before taking a lot, tests at random a sample of 5 motors and accepts the lot if they are all good; otherwise he rejects the lot. Find the probability that:(i) he will accept a lot containing 5defective motors;(ii) he will reject a lot containing only one defective motors.

2. In an engineering examination, a student is considered to have failed, secured second class, first class and distinction according as he scores less than 45%, between 45% and 60%, between 60% and 75% and above 75% respectively. In a particular year 10% of the students failed in the examination and 5% of the students got distinction. Find the percentage of students who have got first class and second class. Given that if

, then and .

3. If 10% of the pens manufactured by the company are defective, find the probability that a box 12 pens contain

a) Exactly two defective pensb) At least two defective pensc) No defective pend) At most two defective pens

4. A letter is known to come either from Calcutta or from Tatanagar. In the half printed postal stamp of the coming states only two consecutive letter are readable. Find the chances of the letter coming from (i) Calcutta (ii) Tatanagar.

5. From a lot of 10 items containing 3 defectives, a sample of 4 items is drawn at random. If the sample is drawn without replacement and the random variable denotes the number of defective items in the sample. Find

a) The probability distribution of

b)

c)

d)

6. In a Normal distribution, 31% of the items are under 45 and 8% are over 64. Find the parameters of the distribution.

7. Prove that Poisson distribution is a limiting case of Binomial distribution. Also evaluate mode of Poisson distribution.

9. Define the properties of Normal distribution. Also prove that mean and mode are equal.

9. The probability distribution of a r.v. is: Determine the constant .

10. The kms in thousands of kms which car owners get with a certain kind of trial is a r.v. having pdf

.

Find the probability that one of these trial will at least kms.

Module 5:

1. In a partially destroyed laboratory record of an analysis of correlation data, the following results only are legible: Variance of x =9, Regression equation: 8x - 10y + 66=0; 40x - 18y - 214=0. Find (i) the mean values of x and y (ii) the S.D. of y (iii) coefficient of correlation between x and y

2. Show that , the acute angle between the two lines of regression, is given by

tanθ=1−r2

r.σ xσ yσx2

+σ y2

.

3. Define the correlation by graphically and mathematically. Also prove that the correlation is independent of change of origin and scale.

4. Obtain the rank correlation coefficient for the following data:X 85 74 85 50 65 78 74 60 74 90Y 78 91 78 58 60 72 80 55 68 70

5. Find the standard deviation of the following data Weekly incme(in Rs.)

20-25 25-30 30-35 35-40 40-45 45-50

no. of workers

200 700 900 800 600 400

6. Evaluate the probabilities at µ =10 and σ = 5 (i) P(X < 15) (ii) P(X > 15) (iii) P(10 ≤ X ≤ 15)

7. The first four moments of a distribution about the value 4 of the variable are -1.5,17,-30 and 108.Find the β1.

8. The first for raw moment is -1.5, 17, -30 and 108 find kurtosis coefficient.

9. Obtain the equation of the line of regression of yield of rice (y) and water (x):Water in inches(x) 12 18 24 30 36 42 48

Yield in tons(y) 5.27 5.68 6.25 7.21 8.02 8.71 8.42

10. Find the rank correlation coefficient from the following:Marks in EVS 43 44 46 40 44 42 45 42 38 40 42 57

Marks in Hindi 29 31 19 18 19 27 27 29 41 30 26 10