Math Formula All Collection

Engineering Mathematics 2013

Prepared by Mr.C.Ganesan, M.Sc.,M.Phil., (Ph: 9841168971) Page 1

NAME OF THE SUBJECT : Engineering Mathematics – I

SUBJECT CODE : MA2111

NAME OF THE METERIAL : Formula Material

MATERIAL CODE : JM08AM1001

(Scan the above Q.R code for the direct download of this material)

Unit – I (Matrices)

1. The Characteristic equation of matrix A is

a) 2

1 20S S if A is 2 X 2 matrix

1

2

Where Sum of the main diagonal elements.S

S A

b) 3 2

1 2 30S S S if A is 3 X 3 matrix

1

2

3

Where Sum of the main diagonal elements.

Sum of the minors of the main diagonal elements.

S

S

S A

2. To find the eigenvectors solve 0A I X .

3. Property of eigenvalues:

Let A be any matrix then

a) Sum of the eigenvalues = Sum of the main diagonal.

b) Product of the eigenvalues = A

c) If the matrix A is triangular then diagonal elements are eigenvalues.

d) If is an eigenvalue of a matrix A, the 1

is the eigenvalue of 1

A .

e) If 1 2, , ...

n are the eigenvalues of a matrix A, then

1 2, , ...

n

m m m are

eigenvalues of mA .( m being a positive integer)

f) The eigenvalues of A & TA are same.

4. Cayley-Hamilton Theorem:

Every square matrix satisfies its own characteristic equation. (ie) 0A I .



5. Matrix of Q.F

1

2

3

2

1 2 1 3

2

2 1 2 3

2

3 1 3 2

1 1( ) ( ) ( )

2 2

1 1( ) ( ) ( )

2 2

1 1( ) ( ) ( )

2 2

coeff x coeff x x coeff x x

coeff x x coeff x coeff x x

coeff x x coeff x x coeff x

6. Index = p = Number of positive eigenvalues

Rank = r = Number of non-zero rows

Signature = s = 2p-r

7. Diagonalisation of a matrix by orthogonal transformation (or) orthogonal

reduction:

Working Rules:

Let A be any square matrix of order n.

Step:1 Find the characteristic equation.

Step:2 Solve the characteristic equation.

Step:3 Find the eigenvectors.

Step:4 Form a normalized model matrix N, such that the eigenvectors are orthogonal.

Step:5 Find TN .

Step:6 Calculate TD=N AN .

Note:

We can apply orthogonal transformation for symmetric matrix only. If any two eigenvalues are equal then we must use a, b, c method for third eigenvector.

Unit – II (Three Dimensional Analytical Geometry)

1. Equation of the sphere, general form 2 2 22 2 2 0x y z ux vy wz d ,

centre

, ,2 2 2

x coefficient y coefficient z coefficient

,

2

x coefficientu

2

y coefficientv

2

z coefficientw

radius 2 2 2r u v w d .



2. Equation of the sphere with centre , ,a b c , radius r is

2 2 2 2

x a y b z c r .

3. Equation of the sphere with centre origin and radius r is 2 2 2 2x y z r .

4. Equation of circle:

The curve of intersection of a sphere by a plane is a circle. So, a circle can be represented by two equations, one being the equation of a sphere and the

other that of a plane. Thus, the equation 2 2 22 2 2 0,x y z ux vy wz d

x my nz p taken together represent a circle.

5. Tangent plane:

Equation of tangent plane of sphere at the point 1 1 1, ,x y z is

1 1 11 1 10u x x v y y w z z dxx yy zz .

6. Condition for the plane x my nz p to be a tangent plane to the sphere

2 2 2 2 2 2 2

u mv nw p m n u v w d .

7. Condition for the spheres to cut orthogonally1 2 1 2 1 2 1 2

2 2 2u u v v w w d d .

8. Equation of Right Circular Cone is

2 2 2 22 2 2 2

cosx m y n z m n x y z

9. Equation of Right Circular Cylinder is

2 2 2

2 2 2 2n y m z z n x m x y r m n

If radius is not given

2

2 2 22

2 2 2

x m y n zr x y z

m n

.

Unit – III (Differential Calculus)

1. Curvature of a circle = Reciprocal of it’s radius

2. Radius of curvature with Cartesian form

3

2 21

2

1 y

y

3. Radius of curvature if1

y ,

3

2 21

2

1 x

x

, where 1

dxx

dy



4. Radius of curvature in implicit form

3

2 22

2 22

x y

xx y xy x y yy x

f f

f f f f f f f

5. Radius of curvature with paramatic form

3

2 2 2x y

x y x y

6. Centre of curvature is ,x y .

7. Circle of curvature is 2 2 2

x x y y .

where 2

1 1

2

1y yx x

y

,

2

1

2

1 yy y

y

8. Evolute: The locus of centre of curvature of the given curve is called evolute of

the curve. 2

1 1

2

1y yx x

y

,

2

1

2

1 yy y

y

9. Envelope: The envelope is a curve which meets each members of a family of

curve.

If the given equation can be rewrite as quadratic equation in parameter, (ie)

20A B C where , , A B C are functions of x and y then the envelope is

24 0B AC .

10. Evolute as the envelope of normals.

Equations Normal equations

24y ax 3

2y xt at at

24x ay 3

2x yt at at

2 2

2 21

x y

a b

2 2

cos sin

ax bya b

2 2

2 21

x y

a b

2 2

sec tan

ax bya b



2 2 2

3 3 3x y a cos sin cos 2x y a

2xy c 2 3c

y xt ctt

Unit – IV (Functions of several variables)

1. Euler’s Theorem:

If f is a homogeneous function of x and y in degree n , then

(i) f f

x y nfx y

(first order)

(ii) 2 2 2

2 2

2 22 1

f f fx xy y n n f

x x y y

(second order)

2. If ( , , )u f x y z , 1 2 3( ), ( ), ( )x g t y g t z g t then

du u dx u dy u dz

dt x dt y dt z dt

3. If 1 2

( , ), ( , ), ( , )u f x y x g r y g r then

(i) u u x u y

r x r y r

(ii)

u u x u y

x y

4. Maxima and Minima :

Working Rules:

Step:1 Find x

f and y

f . Put 0x

f and 0y

f . Find the value of x and y.

Step:2 Calculate , ,xx xy yy

r f s f t f . Now 2rt s

Step:3 i. If 0 , then the function have either maximum or minimum.

1. If 0r Maximum

2. If 0r Minimum

ii. If 0, then the function is neither Maximum nor Minimum, it is

called Saddle Point.

iii. If 0, then the test is inconclusive.

5. Maxima and Minima of a function using Lagrange’s Multipliers:

Let ( , , )f x y z be given function and ( , , )g x y z be the subject to the condition.



Form ( , , ) ( , , ) ( , , )F x y z f x y z g x y z , Putting 0x y z

F F F F and

then find the value of x,y,z. Next we can discuss about the Max. and Min.

6. Jacobian:

Jacobian of two dimensions: , ( , )

, ( , )

u v u vJ

x y x y

u u

x y

v v

x y

7. The functions u and v are called functionally dependent if( , )

0( , )

u v

x y

.

8. ( , ) ( , )

1( , ) ( , )

u v x y

x y u v

9. Taylor’s Expansion:

2 2

3 2 2 3

1 1( , ) ( , ) ( , ) ( , ) ( , ) 2 ( , ) ( , )

1! 2!

1 ( , ) 3 ( , ) 3 ( , ) ( , ) ...

3!

x y xx xy yy

xxx xxy xyy yyy

f x y f a b hf a b kf a b h f a b hkf a b k f a b

h f a b h kf a b hk f a b k f a b

where h x a and k y b

Unit – V (Multiple Integrals)

1. 0

( , )b x

af x y dxdy x : a to b and y : o to x (Here the first integral is w.r.t. y)

2. 0

( , )b y

af x y dxdy x : 0 to y and y : a to b (Here the first integral is w.r.t. x)

3. Area R

dxdy (or) R

dydx

To change the polar coordinate

cos

sin

x r

y r

dxdy rdrd

4. Volume V

dxdydz (or)V

dzdydx

GENERAL:



1. 1

2 2sin

dx x

aa x

(or) 1

2sin

1

dxx

x

2. 2 2

2 2log

dxx a x

a x

(or) 2

2log 1

1

dxx x

x

3. 1

2 2

1tan

dx x

a x a a

(or) 1

2tan

1

dxx

x

4. 2

2 2 2 2 1 sin

2 2

x a xa x dx a x

a

5. /2 /2

0 0

1 3 2sin cos . ... .1 if is odd and 3

2 3

n n n nx dx x dx n n

n n

6. /2 /2

0 0

1 3 1sin cos . ... . if is even

2 2 2

n n n nx dx x dx n

n n


Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: 9841168917) Page 1

SUBJECT NAME : Engineering Mathematics – II

SUBJECT CODE : MA 2161

MATERIAL NAME : Formula Material

MATERIAL CODE : JM08AM1003 (Scan the above Q.R code for the direct download of this material)

Name of the Student: Branch:

Unit – I (Ordinary Differential Equation) 1. ODE with constant coefficients: Solution C.F + P.Iy

Complementary functions:

Sl.No. Nature of Roots C.F

1. 1 2

m m ( )mx

Ax B e

2. 1 2 3

m m m 2 mxAx Bx c e

3. 1 2

m m 1 2m x m xAe Be

4. 1 2 3

m m m 31 2 m xm x m xAe Be Ce

5. 1 2 3

, m m m 3( )m xmx

Ax B e Ce

6. m i ( cos sin )x

e A x B x

7. m i cos sinA x B x

Particular Integral:

Type-I If ( ) 0f x

then . 0P I

Type-II

If ( )ax

f x e

1.

( )

axP I e

D



Replace D by a . If ( ) 0D , then it is P.I. If ( ) 0D , then diff. denominator

w.r.t D and multiply x in numerator. Again replace D by a . If you get denominator

again zero then do the same procedure.

Type-III

Case: i If ( ) sin ( ) cosf x ax or ax

1

. sin (or) cos( )

P I ax axD

Here you have to replace only for 2D not for D . 2

D is replaced by 2a . If the

denominator is equal to zero, then apply same procedure as in Type – I.

Case: ii If 2 2 3 3( ) (or) cos (or) sin (or) cosf x Sin x x x x

Use the following formulas 2 1 cos 2

2

xSin x

, 2 1 cos 2

cos2

xx

,

x x x 3 3 1sin sin sin 3

4 4, x x x 3 3 1

cos cos cos 34 4

and separate 1 2

. & .P I P I

Case: iii If ( ) sin cos ( ) cos sin ( ) cos cos ( ) sin sinf x A B or A B or A B or A B

Use the following formulas:

1( ) in cos ( ) sin( )

2

1(ii) cos sin ( ) sin( )

2

1( ) cos cos cos( ) cos( )

2

1( ) sin sin cos( ) cos( )

2

i s A B sin A B A B

A B Sin A B A B

iii A B A B A B

iv A B A B A B

Type-IV

If ( ) mf x x

1

.( )

mP I xD

1

1 ( )

mxg D



1

1 ( ) mg D x

Here we can use Binomial formula as follows:

i) 1 2 31 1 ...x x x x

ii) 1 2 31 1 ...x x x x

iii) 2 2 31 1 2 3 4 ...x x x x

iv) 2 2 31 1 2 3 4 ...x x x x

v) 3 2 3(1 ) 1 3 6 10 ...x x x x

vi) 3 2 3(1 ) 1 3 6 10 ...x x x x

Type-V

If ( ) axf x e V where sin ,cos , mV ax ax x

1

.( )

axP I e VD

First operate axe by replacing D by D+a.

1

( )

axe VD a

Type-VI

If ( ) nf x x V where sin ,cosV ax ax

sin I.P of

cos R.P of

iax

iax

ax e

ax e

Type-VII (Special Type Problems) If ( ) sec (or) cosec (or) tanf x ax ax ax

1

. ( ) ( )ax axP I f x e e f x dxD a

1. ODE with variable co-efficient: (Euler’s Method)

The equation is of the form 2

2

2( )

d y dyx x y f x

dx dx



Implies that 2 2( 1) ( )x D xD y f x

To convert the variable coefficients into the constant coefficients

Put logz x implies zx e

2 2

3 3

( 1)

( 1)( 2)

xD D

x D D D

x D D D D

where d

Ddx

and d

Ddz

The above equation implies that ( 1) 1 ( )D D D y f x which is O.D.E

with constant coefficients.

2. Legendre’s Linear differential equation:

The equation if of the form 2

2

2( ) ( ) ( )

d y dyax b ax b y f x

dx dx

Put log( )z ax b implies ( ) zax b e

2 2 2

3 3 3

( )

( ) ( 1)

( ) ( 1)( 2)

ax b D aD

ax b D a D D

ax b D a D D D

where d

Ddx

and d

Ddz

3. Method of Variation of Parameters:

The equation is of the form d y dy

a b cy f xdx dx

2

2( )

1 2.C F Ay By and

1 2.P I Py Qy

where 2

1 2 1 2

( )y f xP dx

y y y y

and

1

1 2 1 2

( )y f xQ dx

y y y y

Unit – II (Vector Calculus)

1. Vector differential operator is / / /i x j y k z

2. Gradient of / / /i x j y k z

3. Divergence of F F



4. Curl of F

1 2 3

/ / /

i j k

XF x y z

F F F

5. If F is Solenoidal vector then 0F

6.

7. If F is Irrotational vector the 0XF

8. Maximum Directional derivative

9. Directional derivative of in the direction of a a

a

10. Angle between two normals to the surface 1 2

1 2

cosn n

n n

Where 1 1 1

1 1 ( , , )at x y zn &

2 2 22 2 ( , , )at x y z

n

11. Unit Normal vector, n

12. Equation of tangent plane 1 1 1( ) ( ) ( ) 0l x x m y y n z z

Where l, m,n are coefficient of , ,i j k in .

13. Equation of normal line

1 1 1x x y y z z

l m n

14. Work Done = C

F dr , where dr dxi dyj dzk

15. If .C

F dr be independent of the path is that curl 0F

16. In the surface integral ˆ.

dxdydS

n k ,

ˆ.

dydzdS

n i ,

ˆ.

dzdxdS

n j & ˆdS ndS



17. Green’s Theorem:

If , , ,u v

u vy x

are continuous and one-valued functions in the region R enclosed

by the curve C, then C R

v uudx vdy dxdy

x y

.

18. Stoke’s Theorem:

Let F be the vector point function, around a simple closed curve C and over the

open surface S having as its boundary, then ˆxC S

F dr F nds

19. Gauss Divergence Theorem:

Let F be a vector point function in a region R bounded by a closed surface S,

then ˆS V

F nds Fdv

Unit – III (Analytic Function)

1. Necessary condition for f(z) is analytic function

Cauchy – Riemann Equations: & u v v u

x y x y

(C-R equations)

2. Polar form of Cauchy-Riemann Equations: 1 1

& u v v u

r r r r

3. Condition for Harmonic function: 2 2

2 20

u u

x y

4. If the function is harmonic then it should be either real or imaginary part of a

analytic function.

5. Milne – Thomson method: (To find the analytic function f(z))

i) If u is given ( ) ( ,0) ( ,0)x yf z u z iu z dz

ii) If v is given ( ) ( ,0) ( ,0)y xf z v z iv z dz



6. To find the analytic function

i) ( ) ; ( )f z u iv if z iu v adding these two

We have ( ) ( ) (1 ) ( )u v i u v i f z

then ( )F z U iV where , & ( ) (1 ) ( )U u v V u v F z i f z

Here we can apply Milne – Thomson method for F(z).

7. Bilinear transformation:

1 2 3 1 2 3

1 2 3 1 2 3

w w w w z z z z

w w w w z z z z

Unit – IV (Complex Integration)

1. Cauchy’s Integral Theorem:

If f(z) is analytic and ( )f z is continuous inside and on a simple closed curve C,

then ( ) 0c

f z dz .

2. Cauchy’s Integral Formula:

If f(z) is analytic within and on a simple closed curve C and 0z is any point inside

C, then ( )

2 ( )C

f zdz if a

z a

3. Cauchy’s Integral Formula for derivatives:

If a function f(z) is analytic within and on a simple closed curve C and a is any

point lying in it, then

2

1 ( )( )

2C

f zf a dz

i z a

Similarly

3

2! ( )( )

2C

f zf a dz

i z a

, In general

( )

1

! ( )( )

2

n

n

C

n f zf a dz

i z a

4. Cauchy’s Residue theorem:

If f(z) be analytic at all points inside and on a simple closed cuve c, except for a

finite number of isolated singularities 1 2 3, , ,... nz z z z inside c, then

( ) 2 (sum of the residues of ( ))C

f z dz i f z .



5. Critical point:

The point, at which the mapping w = f(z) is not conformal, (i.e) ( ) 0f z is called

a critical point of the mapping.

6. Fixed points (or) Invariant points:

The fixed points of the transformation az b

wcz d

is obtained by putting w = z in

the above transformation, the point z = a is called fixed point.

7. Re { ( )} ( ) ( )z a

s f z Lt z a f z

(Simple pole)

8. 1

1

1Re { ( )} ( )

( 1)!

mm

mz a

ds f z Lt z a f z

m dz

(Multi Pole (or) Pole of order m)

9. ( )

Re { ( )}( )z a

P zs f z Lt

Q z

10. Taylor Series:

A function ( )f z , analytic inside a circle C with centre at a, can be expanded in

the series

2 3( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ... ( ) ...1! 2! 3! !

nnz a z a z a z a

f z f a f a f a f a f an

Maclaurin’s Series:

Taking a = 0, Taylor’s series reduce to

2 3

( ) (0) (0) (0) (0) ...1! 2! 3!

z z zf z f f f f

11. Laurent’s Series: 0 1

( ) ( )( )

n nn n

n n

bf z a z a

z a

where 1

1

1 ( )

2 ( )n n

C

f za dz

i z a

& 2

1

1 ( )

2 ( )n n

C

f zb dz

i z a

, the integrals being

taken anticlockwise.

12. Isolated Singularity:

A point 0z z is said to be isolated singularity of ( )f z if ( )f z is not analytic at

0z z and there exists a neighborhood of 0z z containing no other singularity.



Example: 1

( )f zz

. This function is analytic everywhere except at 0z .

0z is an isolated singularity.

13. Removable Singularity:

A singular point 0z z is called a removable singularity of ( )f z if

0

lim ( )f z

z z

exists finitely.

Example: sin

lim ( ) lim 1

0 0

zf z

z

z z

(finite) 0z is a removable

singularity.

14. Essential Singularity:

If the principal part contains an infinite number of non zero terms, then 0z z is

known as a essential singularity.

Example:

21 1/1/( ) 1 ...

1! 2!z

zzf z e has 0z as an essential

singularity.

CONTOUR INTEGRATION:

15. Type: I

The integrals of the form 2

0(cos ,sin )f d

Here we shall choose the contour

as the unit circle : 1 or ,0 2iC z z e . On this type 2 1

cos2

z

z

,

2 1sin

2

z

iz

and

1d dz

iz .

16. Type: II

Improper integrals of the form ( )

( )

P xdx

Q x

, where P(x) and Q(x) are polynomials

in x such that the degree of Q exceeds that of P at least by two and Q(x) does not

vanish for any x. Here ( ) ( ) ( )

R

C R

f z dz f x dx f z dz

as ( ) 0R f z dz

.



17. Type: III

The integrals of the form ( ) cos (or) ( )sinf x mxdx f x mxdx

where

( ) 0 as f x x .

Unit – V (Laplace Transform)

1. Definition: 0

( ) ( )stL f t e f t dt

2.

Sl.No

1. 1L 1

s

2. nL t

1 1

! ( 1)n n

n n

s s

3. atL e

1

s a

4. atL e

1

s a

5. sinL at 2 2

a

s a

6. cosL at 2 2

s

s a

7. sinhL at 2 2

a

s a

8. coshL at 2 2

s

s a

3. Linear Property: ( ) ( ) ( ) ( )L af t bg t aL f t bL g t

4. First Shifting property:

If ( ) ( )L f t F s , then

i) ( ) ( )at

s s aL e f t F s

ii) ( ) ( )at

s s aL e f t F s



5. Second Shifting property:

If ( ) ( )L f t F s , ( ),

( )0,

f t a t ag t

t a

then ( ) ( )asL g t e F s

6. Change of scale:

If ( ) ( )L f t F s , then 1

( )s

L f at Fa a

7. Transform of derivative:

If ( ) ( )L f t F s , then ( ) ( )d

L tf t F sds

, 2

2

2( ) ( )

dL t f t F s

ds ,

In general ( ) ( 1) ( )n

n n

n

dL t f t F s

ds

8. Transform of Integral;

If ( ) ( )L f t F s , then 1

( ) ( )s

L f t F s dst

9. Initial value Theorem:

If ( ) ( )L f t F s , then ( ) ( )

0 s

Lt f t Lt sF s

t

10. Final value Theorem:

If ( ) ( )L f t F s , then ( ) ( )

s 0

Lt f t Lt sF s

t

11.

Sl.No

1. 1 1

Ls

1

2. 1 1

Ls a

ate

3. 1 1

Ls a

ate

4. 1

2 2

sL

s a

cos at

5. 1

2 2

1L

s a

1

sin ata



6. 1

2 2

sL

s a

cosh at

7. 1

2 2

1L

s a

1

sinh ata

8. 1 1

nL

s

1

( 1)!

nt

n

12. Deriative of inverse Laplace Transform:

1 11( ) ( )L F s L F s

t

13. Colvolution of two functions: 0

( ) ( ) ( ) ( )

t

f t g t f u g t u du

14. Covolution theorem:

If f(t) & g(t) are functions defined for 0t then ( ) ( ) ( ) ( )L f t g t L f t L g t

15. Convolution theorem of inverse Laplace Transform:

1 1 1( ) ( ) ( ) ( )L F s G s L F s L G s

16. Solving ODE for second order differential equations using Laplace Transform

i) ( ) ( ) (0)L y t sL y t y

ii) 2( ) ( ) (0) (0)L y t s L y t sy y

iii) 3 2( ) ( ) (0) (0) (0)L y t s L y t s y sy y take ( )y L y t

17. Solving integral equation: 0

1( ) ( )

t

L y t dt L y ts

18. Inverse Laplace Transform by Contour Integral method

1 1( ) ( )

2

st

c

L F s F s e dsi

19. Periodic function in Laplace Transform:

If f(x+T) = f(x), then f(x) is periodic function with period T.

0

1( ) ( )

1

Tst

sTL f t e f t dt

e



SUBJECT NAME : Transforms and Partial Diff. Eqns.

SUBJECT CODE : MA2211



Name of the Student: Branch: Unit – I (Fourier Series)

1) Dirichlet’s Conditions:

Any function ( )f x can be expanded as a Fourier

series 0

1 1

cos sin2

n n

n n

aa nx b nx

where 0, ,

n na a b are constants provided the

following conditions are true. ( )f x is periodic, single – valued and finite.

( )f x has a finite number of discontinuities in any one period.

( )f x has at the most a finite number of maxima and minima.

2) The Fourier Series in the interval (0,2π):

0

1 1

( ) cos sin2

n n

n n

af x a nx b nx

Where 2

0

0

1( )a f x dx

,

2

0

1( )cos

na f x nxdx

,

2

0

1( )sin

nb f x nxdx

3) The Fourier Series in the interval (-π,π):

0

1 1

( ) cos sin2

n n

n n

af x a nx b nx

Where 0

0

2( )a f x dx

,

0

2( )cos

na f x nxdx

,

0

2( )sin

nb f x nxdx

In this interval, we have to verify the function is either odd function or

even function. If it is even function then find only 0 and

na a ( 0

nb ). If it is odd

function then find only n

b (0

0n

a a ).



If the function is neither odd nor even then you should find

0, and

n na a b by using the following formulas

0

1( )a f x dx

,

1( )cos

na f x nxdx

, 1

( )sinn

b f x nxdx

.

4) The half range Fourier Series in the interval (0,π):

The half range Cosine Series in the interval (0,π):

0

1

( ) cos2

n

n

af x a nx

Where 0

0

2( )a f x dx

,

0

2( )cos

na f x nxdx

The half range Sine Series in the interval (0,π):

1

( ) sinn

n

f x b nx

Where 0

2( )sin

nb f x nxdx

5) The Parseval’s Identity in the interval (0,2π): 2 2

2 2 20

10

1[ ( )] [ ]

4n n

n

af x dx a b

6) The Parseval’s Identity in the interval (-π,π): 2

2 2 20

10

2[ ( )] [ ]

4n n

n

af x dx a b

7) The Parseval’s Identity for half range cosine series in the interval (0,π): 2

2 20

10

2[ ( )]

4n

n

af x dx a

8) The Parseval’s Identity for half range sine series in the interval (0,π):

2 2

10

2[ ( )]

n

n

f x dx b

Change of interval:

9) The Fourier Series in the interval (0,2ℓ):

0

1 1

( ) cos sin2

n n

n n

a n x n xf x a b

Where 2

0

0

1( )a f x dx ,

2

0

1( )cos

n

n xa f x dx

,

2

0

1( )sin

n

n xb f x dx



10) The Fourier Series in the interval (-ℓ, ℓ):

0

1 1

( ) cos sin2

n n

n n

a n x n xf x a b

Where 0

0

2( )a f x dx ,

0

2( )cos

n

n xa f x dx

,

0

2( )sin

n

n xb f x dx

In this interval, you have to verify the function is either odd function or even

function. If it is even function then find only 0 and

na a ( 0

nb ). If it is odd

function then find only n

b (0

0n

a a ).

11) The half range Fourier Series in the interval (0, ℓ):

The half range Cosine Series in the interval (0, ℓ):

0

1

( ) cos2

n

n

a n xf x a

Where 0

0

2( )a f x dx ,

0

2( )cos

n

n xa f x dx

The half range Sine Series in the interval (0, ℓ):

1

( ) sinn

n

n xf x b

Where 0

2( )sin

n

n xb f x dx

12) The Parseval’s Identity in the interval (0,2ℓ):

2 2

2 2 20

10

1[ ( )] [ ]

4n n

n

af x dx a b

13) The Parseval’s Identity in the interval (-ℓ,ℓ): 2

2 2 20

10

2[ ( )] [ ]

4n n

n

af x dx a b

14) The Parseval’s Identity for half range cosine series in the interval (0,ℓ): 2

2 20

10

2[ ( )]

4n

n

af x dx a

15) The Parseval’s Identity for half range sine series in the interval (0,ℓ):

2 2

10

2[ ( )]

n

n

f x dx b



16) Harmonic Analysis: The method of calculation of Fourier constants by means of numerical

calculation is called as Harmonic analysis.

0

1 1

( ) cos sin2

n n

n n

af x a nx b nx

where

0 1 2 3

2 2 2 2 , cos , cos 2 , cos 3 , ...a y a y x a y x a y x

n n n n

1 2 3

2 2 2sin , b sin 2 , b sin 3 , ...b y x y x y x

n n n

When the values of x is given as numbers the is calculated by 2 x

T

.

Where T is period, n is the number of values given. If the first and last y values are same we can omit one of them.

Complex form of Fourier Series: 17) The Complex form of Fourier Series in the interval (0,2π):

( )inx

n

n

f x c e

where2

0

1( )

2

inx

nc f x e dx

18) The Complex form of Fourier Series in the interval (-π,π):

( )inx

n

n

f x c e

where1

( )2

inx

nc f x e dx

19) The Complex form of Fourier Series in the interval (0,2ℓ):

( )

in x

n

n

f x c e

where2

0

1( )

2

in x

nc f x e dx

20) The Complex form of Fourier Series in the interval (-ℓ,ℓ):

( )

in x

n

n

f x c e

where1

( )2

in x

nc f x e dx

Unit – II (Fourier Transforms)

1) Fourier Integral theorem

The Fourier integral theorem of ( )f x in the interval , is

0

1( ) ( )cos ( )f x f x t dxd



2) Convolution Theorem If [ ]F s and [ ]G s are the Fourier transform of the functions ( )f x and

( )g x respectively, then [ ( )* ( )] .F f x g x F s G s

3) The Fourier Transform of a function ( )f x is given by [ ( )]F f x is denoted by

[ ]F s .

4) Fourier Transform 1

[ ] [ ( )] ( )2

isxF s F f x f x e dx

5) Inverse Fourier Transform 1

( ) [ ]2

isxf x F s e ds

6) The Fourier transforms and Inverse Fourier transforms are called Fourier transforms pairs.

7) Fourier Sine Transform 0

2[ ] [ ( )] ( )sin

s sF s F f x f x sx dx

8) Fourier Cosine Transform 0

2[ ] [ ( )] ( )cos

c cF s F f x f x sx dx

9) If ( )ax

f x e then the Fourier Cosine and Sine transforms as follows

a) 2 2

2[ ( )]

c

aF f x

a s

b) 2 2

2[ ( )]

s

sF f x

a s

10) Property

a) [ ( )] [ ( )]s c

dF xf x F f x

ds

b) [ ( )] [ ( )]c s

dF xf x F f x

ds

11) Parseval’s Identity

a) 2 2

( ) ( )F s ds f x dx

b) 0 0

( ) ( ) ( ) ( )c c

F s G s ds f x g x dx

(Or) 2 2

0 0

( ) ( )c

F s ds f x dx

12) Condition for Self reciprocal [ ( )] ( )F f x f s



Unit – III (Partial Differential Equation) 1) Lagrange’s Linear equation

The equation of the form Pp Qq R

then the subsidiary equation is dx dy dz

P Q R

2) Homogeneous Linear Partial Differential Equation of higher order with constant

coefficients:

The equation of the form 2 2 2

2 2( , )

z z za b c f x y

x x y y

The above equation can be written as

2 2( , )aD bDD cD z f x y …………………………….. (1)

where2

2

2 , D D

x x

and 2

2

2 , D D

y y

The solution of above equation is z = C.F + P.I Complementary Function (C.F) :

To find C.F consider the auxiliary equation by replacing D by m and D by

1.The equation (1) implies that 20am bm c , solving this equation

we get two values of m. The following table gives C.F of the above equation.

Sl.No. Nature of m Complementary Function

1 1 2m m

1 1 2 2C.F = ( ) ( )f y m x f y m x

2 1 2m m

1 2C.F = ( ) ( )f y mx xf y mx

3 1 2 3m m m

1 1 2 2 3 3C.F = ( ) ( ) ( )f y m x f y m x f y m x

4 1 2 3m m m 2

1 2 3C.F = ( ) ( ) ( )f y mx xf y mx x f y mx

5 1 2 3, is differentm m m

1 2 3 3C.F = ( ) ( ) ( )f y mx xf y mx f y m x

Particular Integral (P.I) :

To find P.I consider 2 2( , )D D aD bDD cD .

Type: 1 If ( , ) 0f x y , then P.I 0 .

Type: 2 If ( , )ax by

f x y e

1.

( , )

ax byP I e

D D



Replace D by a and D by b. If ( , ) 0D D , then it is P.I.

If ( , ) 0D D , then diff. denominator w.r.t D and multiply x in

numerator. Again replace D by a and D by b. If again denominator equal to zero then continue the same procedure.

Type: 3 If ( , ) sin( ) (o ) cos( )f x y ax by r ax by

1. sin( ) (o ) cos( )

( , )P I ax by r ax by

D D

Here replace 2D by 2

a , 2D by 2

b and DD by ab . Do not replace for and D D . If the denominator equal to zero, then apply the same producer as in Type: 2.

Type: 4 If ( , )m n

f x y x y

1.

( , )

m nP I x y

D D

1

1 ( , )

m nx y

g D D

1

1 ( , )m n

g D D x y

Here we can use Binomial formula as follows:

i) 1 2 3

1 1 ...x x x x

ii) 1 2 3

1 1 ...x x x x

iii) 2 2 3

1 1 2 3 4 ...x x x x

iv) 2 2 3

1 1 2 3 4 ...x x x x

v) 3 2 3(1 ) 1 3 6 10 ...x x x x

vi) 3 2 3(1 ) 1 3 6 10 ...x x x x

Type: 5 If ( , ) Vax by

f x y e , where

V=sin( ) (or) cos( ) (or) m n

ax by ax by x y

1. V

( , )

ax byP I e

D D

First operate ax bye

by replacing D by D a and Dby D a .

1. V

( , )

ax byP I e

D a D b

, Now this will either Type: 3 or

Type: 4.



Type: 6 If ( , ) sin (or) cosf x y y ax y ax

1. sin

( , )P I y ax

D D

1 2

1 siny ax

D m D D m D

2y c m x

2

1

1 sin c m x ax dx

D m D

(Apply Bernouili’s method)

3) Solution of Partial Differential Equations:

Standard Type: 1 Equation of the form ( , ) 0f p q

Assume that z ax by c be the solution the above

equation.put and p a q b in equation (1), we get

( , ) 0f a b . Now, solve this, we get ( )b a .

( )z ax a y c which is called Complete solution.

Standard Type: 2 Equation of the form ( , )z px qy f p q (Clairaut’s form)

The Complete solution is ( , )z ax by f a b . To find

Singular integral diff. partially w.r.t a & b , equate to zero

and eliminate a and b .

Standard Type: 3 Equation of the form 1 2( , ) ( , )f x p f y q

The solution is z pdx qdy .

Standard Type: 4 Equation of the form ( , , ) 0f z p q

In this type put u x ay , then ,dz dz

p q adu du

Unit – IV (Application of Partial Differential Equation) 1) The One dimensional Wave equation:

2 2

2

2 2

y ya

t x

The three solutions of the above equation are

i) ( , )px px pat pat

y x t Ae Be Ce De

ii) ( , ) cos sin cos siny x t A px B px C pat D pat

iii) ( , )y x t Ax B Ct D



But the correct solution is ii),

( , ) cos sin cos siny x t A px B px C pat D pat .

2) The One dimensional Heat flow equation: 2

2

2

u u

t x

2 k

c

where

Thermal Conductivity

Density

Specific Heat

k

c


i) 2 2

( , )px px p t

u x t Ae Be Ce

ii) 2 2

( , ) cos sinp t

u x t A px B px Ce

iii) ( , )u x t Ax B C

But the correct solution is ii), 2 2

( , ) cos sinp t

u x t A px B px Ce

3) The Two dimensional Heat flow equation: 2 2

2 20

u u

x y


i) ( , ) cos sinpx px

u x y Ae Be C py D py

(Applicable when given value is parallel to y-axies)

ii) ( , ) cos sinpy py

u x y A px B px Ce De

(Applicable when given value is parallel to x-axies)

iii) ( , )u x y Ax B Cy D (Not applicable)

Unit – V (Z - Transform)

1) Definition of Z-transform:

Let ( )f n be the sequence defined for all the positive integers n such that

0

( ) ( )n

n

Z f n f n z



2)

Sl.No ( )Z f n [ ]F z

1. 1Z 1

z

z

2. ( 1)n

Z 1

z

z

3. n

Z a z

z a

4. Z n

21

z

z

5. 1Z n

2

21

z

z

6. 1

Zn

log1

z

z

7. sin2

nZ

2

1

z

z

8. cos2

nZ

2

21

z

z

3) Statement of Initial value theorem:

If ( ) [ ]Z f n F z , then 0

[ ] ( )z nLt F z Lt f n

4) Statement of Final value theorem:

If ( ) [ ]Z f n F z , then 1

( ) ( 1) ( )n zLt f n Lt z F z

5) ( ) ( )n

zz

a

Z a f n Z f n

6) ( ) ( )d

Z nf n z Z f ndz

7) Inverse Z-transform

Sl.No 1( )Z F z

( )f n

1. 1

1

zZ

z

1

2. 1

1

zZ

z

( 1)

n

3. 1 z

Zz a

n

a



4. 1 z

Zz a

na

5.

1

21

zZ

z

n

6.

1

2

zZ

z a

1nna

7.

1

2

zZ

z a

1n

n a

8.

2

1

21

zZ

z

cos

2

n

9.

2

1

2 2

zZ

z a

cos

2

n na

10. 1

21

zZ

z

sin2

n

11. 1

2 2

zZ

z a

1

sin2

n na

8) Inverse form of Convolution Theorem

1 1 1[ ( ). ( )] [ ( )] [ ( )]Z F z G z Z F z Z G z

and by the defn. of Convolution of two functions 0

( ) ( ) ( ) ( )n

r

f n g n f r g n r

9) a) [ ( )] ( )Z y n F z

b) [ ( 1)] ( ) (0)Z y n zF z zy

c) 2 2[ ( 2)] ( ) (0) (1)Z y n z F z z y zy

d) 3 3 2[ ( 3)] ( ) (0) (1) (2)Z y n z F z z y z y zy

----All the Best----



SUBJECT NAME : Numerical Methods





UNIT – 1

1. Regula Falsi method: 1

( ) ( )

( ) ( )

af b bf ax

f b f a

2. Newton’s method (0r) Newton-Raphson method:1 /

( )

( )n

n n

n

f xx x

f x

3. Fixed point iteration (or) Iterative formula (or) Simple iteration method

( )1

x g xn n

, ( )where x g x

4. The rate of convergence in N-R method is of order 2.

5. Condition for convergence of N-R method is 2// /( ) ( ) ( )f x f x f x

6. Condition for the convergence of iteration method is /( ) 1g x

7. Gauss elimination & Gauss-Jordon are direct methods and Gauss-Seidal and

Gauss-Jacobi are iterative methods.

8. Power method, To find numerically largest eigen value 1n nY AX

UNIT – 2

1. Lagrange’s interpolation formula is

0 2 0 1 11 20 1

0 1 0 2 0 1 0 1 2 1 0 1 1

( )( )...( ) ( )( )...( )( )( )...( )( ) ...

( )( )...( ) ( )( )...( ) ( )( )...( )n nn

n

n n n n n n

x x x x x x x x x x x xx x x x x xy f x y y y

x x x x x x x x x x x x x x x x x x

2. Inverse of Lagrange’s interpolation formula is

0 2 0 1 11 20 1

0 1 0 2 0 1 0 1 2 1 0 1 1

( )( )...( ) ( )( )...( )( )( )...( )( ) ...

( )( )...( ) ( )( )...( ) ( )( )...( )n nn

n

n n n n n n

y y y y y y y y y y y yy y y y y yx f y x x x

y y y y y y y y y y y y y y y y y y

3. Newton’s divided difference interpolation formula is 2

( ) ( ) ( ) ( ) ( )( ) ( ) ...0 0 0 0 1 0f x f x x x f x x x x x f x

4. Newton’s forward difference formula is

2 3

0 0 0 0

( 1) ( 1)( 2)( ) ...

1! 2! 3!

u u u u u uy x y y y y



where 0x xu

h

5. Newton’s backward difference formula is

2 3( 1) ( 1)( 2)( ) ...

1! 2! 3!n n n n

v v v v v vy x y y y y

where x xnvh

6. Interpolation with a cubic spline

1 1 1 12

64 2i i i i i iM M M y y y

h

23 3

1 1 1 1

1 1( ) ( ) ( ) ( )

6 6i i i i i i i

hS x x x M x x M x x y M

h h

2

1

1( )

6i i i

hx x y M

h

1,2,...( 1) and xi-1for i n x xi

UNIT – 3

1. Newton’s forward formula to find the derivatives 2

2 3

0 0 0

1 2 1 3 6 2...

2 6

dy u u uy y y

dx h

2 22 3 4

0 0 02 2

1 6 18 11( 1) ...

12

d y u uy u y y

dx h

3

3 4

0 03 3

1 12 18...

12

d y uy y

dx h

where 0x x

uh

2. Newton’s forward formula to find the derivatives at 0x x

2 3

0 00

0

1...

2 3X X

y ydyy

dx h

22 3 4

0 0 02 2

0

1 11...

12X X

d yy y y

dx h

33 4

0 03 3

0

1 3...

2X X

d yy y

dx h

3. Newton’s backward formula to find the derivatives 2

2 31 2 1 3 6 2...

2 6n n n

dy v v vy y y

dx h

2 22 3 4

2 2

1 6 18 11( 1) ...

12n n n

d y v vy v y y

dx h



33 4

3 3

1 12 18...

12n n

d y vy y

dx h

where nx x

vh

4. Newton’s backward formula to find the derivatives at nx x

2 31...

2 3n n

n

X Xn

dy y yy

dx h

22 3 4

2 2

1 11...

12n n n

X Xn

d yy y y

dx h

33 4

3 3

1 3...

2n n

X Xn

d yy y

dx h

5. Trapezoidal Rule

0

0 1 2 3

0

( ) 2 ...2

x nh

n

x

hf x dx y y y y y

6. Simpson’s 1/3rd Rule

0

0 1 3 5 2 4 6

0

( ) 4 ... 2 ...3

x nh

n

x

hf x dx y y y y y y y y

7. Simpson’s 3/8th Rule

0

0 1 2 4 5 7 3 6 9

0

3( ) 3 ... 2 ...

8

x nh

n

x

hf x dx y y y y y y y y y y

8. Romberg’s formula for 1 2 & is

2 12

3

9. Error is the Trapezoidal formula is of the order 2h and in the Simpson formula is

of the order4h .

10. Two points Gaussian Quadrature formula 1

1

1 1( )

3 3f x dx f f

11. Three points Gaussian Quadrature formula 1

1

5 3 3 8( ) (0)

9 5 5 9f x dx f f f

12. If the range is not 1 ,1 then the idea to solve the Gaussian Quadrature

problem is 2 2

b a b ax z

UNIT – 4



1. Taylor Series method 2 3

/ // ///

1 0 0 0 0 0( ) ...1! 2! 3!

h h hy y x h y y y y

and 2 3

/ // ///

2 1 1 1 1 1( ) ...1! 2! 3!

h h hy y x h y y y y

2. Euler’s method

If 0 0( , ); ( )

dyf x y y x y

dx

then 0 0 0 0( ) ( , )y x h y hf x y

and 1 1 1 1( ) ( , )y x h y hf x y

3. Modified Euler’s method

If 0 0( , ); ( )

dyf x y y x y

dx

then ( ) , ( , )0 0 0 00 0 2 2 h h

y x h y h x y f x yf

And ( ) , ( , )1 1 1 11 1 2 2 h h

y x h y h x y f x yf

4. Runge-kutta method of fourth order

First formula:

1 0 0( , )k hf x y 23 0 0,

2 2

h kk hf x y

12 0 0,

2 2

h kk hf x y

4 0 0 3,k hf x h y k

and 1 2 3 4

12 2

6y k k k k & 0 0( )y x h y y

For the second formula, replace 0 by 1 in the above formula.

5. Milne’s Predictor formula: / / /

1 3 2 1

42 2

3n n n n n

hy y y y y

6. Milne’s Corrector formula: / / /

1 1 1 143

n n n n n

hy y y y y

7. Adams-Bashforth method:

Adam’s Predictor formula:/ / / /

1 1 2 355 59 37 924

n n n n n n

hy y y y y y

Adam’s Corrector formula:/ / / /

1 1 1 29 19 524

n n n n n n

hy y y y y y

UNIT – 5



1. Solving ordinary diff. Eqn. by finite diff. method

The given eqn. // /( ) ( ) ( ) ( ) ( ) ( )y x f x y x g x y x x

// 1 12

2( ) i i i

i

y y yy x

h

& / 1 1( )

2i i

i

y yy x

h

2. Solving one dimensional heat eqn. by Bender –Schmidt’s method [Explicit

method]

The given equation

2

2

u ua

x t

then , 1 1, , 1,(1 2 )i j i j i j i ju u u u , where 2

k

ah

for 1

2 the above equation becomes,

, 1 1, 1,

1

2i j i j i ju u u

, where 2

2

ak h

3. Solving one dimensional heat eqn. by Crank-Nicolson’s method [Implicit

method]

The given equation

2

2

u ua

x t

then 1, 1 1, 1 , 1 , 1, 1,2( 1) 2( 1) ( )i j i j i j i j i j i ju u u u u u

where 2

k

ah

for 1 the above equation becomes,

, 1 1, 1 1, 1 1, 1,

1

4i j i j i j i j i ju u u u u

, where 2k ah

4. Solving one dimensional wave eqn.

The given equation

2 22

2 2

u ua

t x

then 2 2 2 2

, 1 , 1, 1, , 12 1 ( )i j i j i j i j i ju a u a u u u ,

where k

h

for 2 2 1a , the above equation becomes,



, 1 1, 1, , 1i j i j i j i ju u u u , where h

ka

5. Solving two dimensional Laplace equation

The given eqn.

2 2

2 20

u u

x y

(or)

2 0u

The Standard five point formula: [SFPF]

, 1, 1, , 1 , 1

1


Diagonal five point formula: [DFPF]

, 1, 1 1, 1 1, 1 1, 1

1


Liebamann’s iteration process:

( 1) ( 1) ( ) ( ) ( 1)

, 1, 1, , 1 , 1

1

4

n n n n n

i j i j i j i j i ju u u u u

6. Solving two dimensional Poisson equation

The given eqn.

2 2

2 2( , )

u uf x y

x y

(or)

2 ( , )u f x y

To solve the above equation, we use the following formula

2

1, 1, , 1 , 1 ,4 ( , )i j i j i j i j i ju u u u u h f ih jh

---- All the Best ----



SUBJECT NAME : Probability & Queueing Theory





UNIT-I (RANDOM VARIABLES)

1) Discrete random variable: A random variable whose set of possible values is either finite or countably infinite is called discrete random variable. Eg: (i) Let X represent the sum of the numbers on the 2 dice, when two dice are thrown. In this case the random variable X takes the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. So X is a discrete random variable. (ii) Number of transmitted bits received in error.

2) Continuous random variable: A random variable X is said to be continuous if it takes all possible values between certain limits. Eg: The length of time during which a vacuum tube installed in a circuit functions is a continuous random variable, number of scratches on a surface, proportion of defective parts among 1000 tested, number of transmitted in error.

3)

Sl.No. Discrete random variable Continuous random variable 1

( ) 1i

i

p x

( ) 1f x dx

2 ( )F x P X x ( ) ( )

x

F x P X x f x dx

3 Mean ( )i i

i

E X x p x Mean ( )E X xf x dx

4 2 2( )

i i

i

E X x p x 2 2( )E X x f x dx

5 2

2Var X E X E X

22

Var X E X E X

6 Moment = r r

i i

i

E X x p Moment = ( )

r rE X x f x dx



7 M.G.F

( )tX tx

X

x

M t E e e p x

M.G.F

( )tX tx

XM t E e e f x dx

4) E aX b aE X b

5) 2Var VaraX b a X

6) 2 2Var VaraX bY a X b Var Y

7) Standard Deviation Var X

8) ( ) ( )f x F x 9) ( ) 1 ( )p X a p X a

10)

/p A B

p A Bp B

, 0p B

11) If A and B are independent, then p A B p A p B .

12) 1st Moment about origin = E X = 0

Xt

M t

(Mean)

2nd Moment about origin = 2E X =

0X

t

M t

The co-efficient of !

rt

r = r

E X (rth Moment about the origin)

13) Limitation of M.G.F: i) A random variable X may have no moments although its m.g.f exists.

ii) A random variable X can have its m.g.f and some or all moments, yet the m.g.f does not generate the moments.

iii) A random variable X can have all or some moments, but m.g.f does not exist except perhaps at one point.

14) Properties of M.G.F: i) If Y = aX + b, then bt

Y XM t e M at .

ii) cX XM t M ct , where c is constant.

iii) If X and Y are two independent random variables then

X Y X YM t M t M t

.

15) P.D.F, M.G.F, Mean and Variance of all the distributions: Sl.No.

Distribution

P.D.F ( ( )P X x ) M.G.F Mean Variance

1 Binomial x n x

xnc p q

n

tq pe

np npq

2 Poisson

!

xe

x

1

te

e



3 Geometric 1xq p

(or) xq p

1

t

t

pe

qe

1

p

2

q

p

4 Negative Binomial 1

( 1)k x

kx k C p p

1

k

t

p

qe

kq

p

2

kq

p

5 Uniform 1

, ( )

0, otherwise

a x bf x b a

( )

bt ate e

b a t

2

a b

2( )

12

b a

6 Exponential

, 0, 0( )

0, otherwise

xe x

f x

t

1

2

1

7 Gamma 1

( ) , 0 , 0( )

xe x

f x x

1

(1 )t

8 Weibull 1( ) , 0, , 0

xf x x e x

16) Memoryless property of exponential distribution

/P X S t X S P X t .

UNIT-II (RANDOM VARIABLES)

1) 1ij

i j

p (Discrete random variable)

( , ) 1f x y dxdy

(Continuous random variable)

2) Conditional probability function X given Y, ,

/( )

i i

P x yP X x Y y

P y .

Conditional probability function Y given X , ,

/( )

i i

P x yP Y y X x

P x .

,

/( )

P X a Y bP X a Y b

P Y b

3) Conditional density function of X given Y, ( , )

( / )( )

f x yf x y

f y .

Conditional density function of Y given X, ( , )

( / )( )

f x yf y x

f x .



4) If X and Y are independent random variables then

( , ) ( ). ( )f x y f x f y (for continuous random variable)

, .P X x Y y P X x P Y y (for discrete random variable)

5) Joint probability density function , ( , )

d b

c a

P a X b c Y d f x y dxdy .

0 0

, ( , )

b a

P X a Y b f x y dxdy

6) Marginal density function of X, ( ) ( ) ( , )X

f x f x f x y dy

Marginal density function of Y, ( ) ( ) ( , )Y

f y f y f x y dx

7) ( 1) 1 ( 1)P X Y P X Y

8) Correlation co – efficient (Discrete): ( , )

( , )X Y

Cov X Yx y

1( , )Cov X Y XY XY

n , 2 21

XX X

n , 2 21

YY Y

n

9) Correlation co – efficient (Continuous): ( , )

( , )X Y

Cov X Yx y

( , ) ,Cov X Y E X Y E X E Y , ( )X

Var X , ( )Y

Var Y

10) If X and Y are uncorrelated random variables, then ( , ) 0Cov X Y .

11) ( )E X xf x dx

, ( )E Y yf y dy

, , ( , )E X Y xyf x y dxdy

.

12) Regression for Discrete random variable:

Regression line X on Y is xyx x b y y ,

2xy

x x y yb

y y



Regression line Y on X is yxy y b x x ,

2yx

x x y yb

x x

Correlation through the regression, .XY YX

b b Note: ( , ) ( , )x y r x y

13) Regression for Continuous random variable:

Regression line X on Y is ( ) ( )xy

x E x b y E y , x

xy

y

b r

Regression line Y on X is ( ) ( )yx

y E y b x E x , y

yx

x

b r

Regression curve X on Y is / / x E x y x f x y dx

Regression curve Y on X is / / y E y x y f y x dy

14) Transformation Random Variables:

( ) ( )Y X

dxf y f x

dy (One dimensional random variable)

( , ) ( , )UV XY

x x

u vf u v f x y

y y

u v

(Two dimensional random variable)

15) Central limit theorem (Liapounoff’s form)

If X1, X2, …Xn be a sequence of independent R.Vs with E[Xi] = µi and Var(Xi) = σi2, i

= 1,2,…n and if Sn = X1 + X2 + … + Xn then under certain general conditions, Sn

follows a normal distribution with mean 1

n

i

i

and variance 2 2

1

n

i

i

as

n .

16) Central limit theorem (Lindberg – Levy’s form)

If X1, X2, …Xn be a sequence of independent identically distributed R.Vs with E[Xi]

= µi and Var(Xi) = σi2, i = 1,2,…n and if Sn = X1 + X2 + … + Xn then under certain



general conditions, Sn follows a normal distribution with mean n and variance

2n as n .

Note: nS n

zn

( for n variables),

Xz

n

( for single variables)

UNIT-III (MARKOV PROCESSES AND MARKOV CHAINS)

1) Random Process:

A random process is a collection of random variables {X(s,t)} that are

functions of a real variable, namely time ‘t’ where s Є S and t Є T.

2) Classification of Random Processes:

We can classify the random process according to the characteristics of time t

and the random variable X. We shall consider only four cases based on t and X

having values in the ranges -∞< t <∞ and -∞ < x < ∞.

Continuous random process

Continuous random sequence

Discrete random process

Discrete random sequence

Continuous random process:

If X and t are continuous, then we call X(t) , a Continuous Random Process.

Example: If X(t) represents the maximum temperature at a place in the

interval (0,t), {X(t)} is a Continuous Random Process.

Continuous Random Sequence:

A random process for which X is continuous but time takes only discrete values is

called a Continuous Random Sequence.

Example: If Xn represents the temperature at the end of the nth hour of a day, then

{Xn, 1≤n≤24} is a Continuous Random Sequence.

Discrete Random Process:

If X assumes only discrete values and t is continuous, then we call such random

process {X(t)} as Discrete Random Process.

Example: If X(t) represents the number of telephone calls received in the interval

(0,t) the {X(t)} is a discrete random process since S = {0,1,2,3, . . . }



Discrete Random Sequence:

A random process in which both the random variable and time are discrete is

called Discrete Random Sequence.

Example: If Xn represents the outcome of the nth toss of a fair die, the {Xn : n≥1} is a

discrete random sequence. Since T = {1,2,3, . . . } and S = {1,2,3,4,5,6}

3) Condition for Stationary Process: ( ) ConstantE X t , ( ) constantVar X t .

If the process is not stationary then it is called evolutionary.

4) Wide Sense Stationary (or) Weak Sense Stationary (or) Covariance Stationary: A random process is said to be WSS or Covariance Stationary if it satisfies the

following conditions.

i) The mean of the process is constant (i.e) ( ) constantE X t .

ii) Auto correlation function depends only on (i.e)

( ) ( ). ( )XX

R E X t X t

5) Property of autocorrelation:

(i) 2

( ) lim

XXE X t R

(ii) 2( ) 0

XXE X t R

6) Markov process: A random process in which the future value depends only on the present value and not on the past values, is called a markov process. It is symbolically

represented by 1 1 1 1 0 0

( ) / ( ) , ( ) ... ( )n n n n n n

P X t x X t x X t x X t x

1 1( ) / ( )

n n n nP X t x X t x

Where 0 1 2 1

...n n

t t t t t

7) Markov Chain:

If for all n , 1 1 2 2 0 0

/ , , ...n n n n n n

P X a X a X a X a

1 1/

n n n nP X a X a

then the process n

X , 0,1, 2, ...n is called the

markov chain. Where 0 1 2, , , ... , ...

na a a a are called the states of the markov chain.

8) Transition Probability Matrix (tpm): When the Markov Chain is homogenous, the one step transition probability is denoted by Pij. The matrix P = {Pij} is called transition probability matrix.

9) Chapman – Kolmogorov theorem: If ‘P’ is the tpm of a homogeneous Markov chain, then the n – step tpm P(n) is

equal to Pn. (i.e) ( )n

n

ij ijP P .

10) Markov Chain property: If 1 2 3, , , then P and

1 2 31 .



11) Poisson process: If ( )X t represents the number of occurrences of a certain event in (0, )t ,then

the discrete random process ( )X t is called the Poisson process, provided the

following postulates are satisfied.

(i) 1 occurrence in ( , )P t t t t O t

(ii) 0 occurrence in ( , ) 1P t t t t O t

(iii) 2 or more occurrences in ( , )P t t t O t

(iv) ( )X t is independent of the number of occurrences of the event in any

interval.

12) Probability law of Poisson process:

( ) , 0,1, 2, ...!

nte t

P X t n nn

Mean ( )E X t t , 2 2 2( )E X t t t , ( )Var X t t .

UNIT-IV (QUEUEING THEORY)

n – Number of customers in the system.

– Mean arrival rate.

– Mean service rate.

nP – Steady State probability of exactly n customers in the system.

qL – Average number of customers in the queue.

sL – Average number of customers in the system.

qW – Average waiting time per customer in the queue.

sW – Average waiting time per customer in the system.

Model – I (M / M / 1): (∞ / FIFO)

1) Server Utilization

2) 1n

nP (P0 no customers in the system)

3) 1

sL



4) 2

1q

L

5)

1

1s

W

6) 1

qW

7) Probability that the waiting time of a customer in the system exceeds t is

( )( )

t

sP w t e

.

8) Probability that the quue size exceeds “t” is 1nP N n where 1n t

.

Model – II (M / M / C): (∞ / FIFO)

1) s

2)

11

0

0 ! ! 1

n ss

n

s sP

n s

3)

1

02

1

. ! 1

s

q

sL P

s s

4) s q

L L s

5) q

q

LW

6) s

s

LW

7) The probability that an arrival has to wait: 0

! 1

ss

P N s Ps

8) The probability that an arrival enters the service without waiting = 1 – P(an

arrival hat to wait) = 1 P N s

9) ( 1 )

0

( ) 11

!(1 )( 1 )

s t s s

ts e

P w t e Ps s s



Model – III (M / M / 1): (K / FIFO)

1)

2) 0 1

1

1k

P

(No customer)

3) 01 P (effective arrival rate)

4) 1

1

1

1 1

k

s k

kL

5) q s

L L

6) s

s

LW

7) q

q

LW

8) 0a customer turned away

k

kP P P

Model – IV (M / M / C): (K / FIFO)

1) s

2)

11

0

0 ! !

n ss k

n s

n n s

s sP

n s

3)

0

0

, !

, !

n

n n

n s

sP n s

nP

sP s n k

s s

4) Effective arrival rate: 1

0

s

n

n

s s n P

5)

1

02

1

! 11

s k s k s

q

s k sL P

s



6) s q

L L

7) q

q

LW

8) s

s

LW

UNIT-V (NON – MARKOVIAN & QUEUEING NETWORK)

1) Pollaczek – Khintchine formula:

22( ) ( )

( )2 1 ( )

S

Var t E tL E t

E t

(or)

2 2 2

2 1S

L

2) Little’s formulas:

2 2 2

2 1S

L

q SL L

S

S

LW

q

q

LW

3) Series queue (or) Tandem queue:

The balance equation

00 2 01P P

1 10 00 2 11P P P

01 2 01 1 10 2 1bP P P P

1 11 2 11 01P P P

2 1 1 11bP P



Condition 00 10 01 11 1

1b

P P P P P

4) Open Jackson networks:

i) Jackson’s flow balance equation 1

k

j j i ij

i

r P

Where k – number of nodes, rj – customers from outside

ii) Joint steady state probabilities

1 2

1 2 1 1 2 2, , ... 1 1 ... 1knn n

k k kP n n n

iii) Average number of customers in the system

1 2

1 2

...1 1 1

k

S

k

L

iv) Average waiting time of a customers in the system

S

S

LW

where

1 2...

kr r r

5) Closed Jackson networks:

In the closed network, there are no customers from outside, therefore 0j

r

then

i) The Jackson’s flow balance equation 1

k

j i ij

i

P

0j

r

(or)

11 12 1

221 22

1 2 1 2

1 2

...

... ... ...

...

k

k

k k

k k kk

P P P

PP P

P P P

ii) If each nodes single server

1 2

1 2 1 2, , ... ... knn n

k N kP n n n C

Where 1 2

1 2

1

1 2

...

... k

k

nn n

N k

n n n N

C

iii) If each nodes has multiple servers



1 2

1 2

1 2

1 2

, , ... ...knn n

k

k N

k

P n n n Ca a a

Where 1 2

1 2

1 1 2

... 1 2

...k

k

nn n

k

N

n n n N k

Ca a a

! ,

! ,i i

i i i

i n s

i i i i

n n sa

s s n s




SUBJECT NAME : Probability & Random Process





UNIT-I (RANDOM VARIABLES)

1) Discrete random variable: A random variable whose set of possible values is either finite or countably infinite is called discrete random variable. Eg: (i) Let X represent the sum of the numbers on the 2 dice, when two dice are thrown. In this case the random variable X takes the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. So X is a discrete random variable. (ii) Number of transmitted bits received in error.

2) Continuous random variable: A random variable X is said to be continuous if it takes all possible values between certain limits. Eg: The length of time during which a vacuum tube installed in a circuit functions is a continuous random variable, number of scratches on a surface, proportion of defective parts among 1000 tested, number of transmitted in error.

3)

Sl.No. Discrete random variable Continuous random variable 1

( ) 1i

i

p x

( ) 1f x dx

2 ( )F x P X x ( ) ( )

x

F x P X x f x dx

3 Mean ( )i i

i

E X x p x Mean ( )E X xf x dx

4 2 2( )

i i

i

E X x p x 2 2( )E X x f x dx

5 2

2Var X E X E X

22

Var X E X E X

6 Moment = r r

i i

i

E X x p Moment = ( )

r rE X x f x dx



7 M.G.F

( )tX tx

X

x

M t E e e p x

M.G.F

( )tX tx

XM t E e e f x dx

4) E aX b aE X b

5) 2Var VaraX b a X

6) 2 2Var VaraX bY a X b Var Y

7) Standard Deviation Var X

8) ( ) ( )f x F x 9) ( ) 1 ( )p X a p X a

10)

/p A B

p A Bp B

, 0p B

11) If A and B are independent, then p A B p A p B .

12) 1st Moment about origin = E X = 0

Xt

M t

(Mean)

2nd Moment about origin = 2E X =

0X

t

M t

The co-efficient of !

rt

r = r

E X (rth Moment about the origin)

13) Limitation of M.G.F: i) A random variable X may have no moments although its m.g.f exists.

ii) A random variable X can have its m.g.f and some or all moments, yet the m.g.f does not generate the moments.

iii) A random variable X can have all or some moments, but m.g.f does not exist except perhaps at one point.

14) Properties of M.G.F: i) If Y = aX + b, then bt

Y XM t e M at .

ii) cX XM t M ct , where c is constant.

iii) If X and Y are two independent random variables then

X Y X YM t M t M t

.

15) P.D.F, M.G.F, Mean and Variance of all the distributions: Sl.No.

Distribution

P.D.F ( ( )P X x ) M.G.F Mean Variance

1 Binomial x n x

xnc p q

n

tq pe

np npq

2 Poisson

!

xe

x

1

te

e



3 Geometric 1xq p

(or) xq p

1

t

t

pe

qe

1

p

2

q

p

4 Uniform 1

, ( )

0, otherwise

a x bf x b a

( )

bt ate e

b a t

2

a b

2( )

12

b a

5 Exponential

, 0, 0( )

0, otherwise

xe x

f x

t

1

2

1

6 Gamma 1

( ) , 0 , 0( )

xe x

f x x

1

(1 )t

7 Normal 21

21( )

2

x

f x e

2 2

2

tt

e

2

16) Memoryless property of exponential distribution

/P X S t X S P X t .

17) Function of random variable: ( ) ( )Y X

dxf y f x

dy

UNIT-II (RANDOM VARIABLES)

1) 1ij

i j

p (Discrete random variable)

( , ) 1f x y dxdy

(Continuous random variable)

2) Conditional probability function X given Y ,

/( )

i i

P x yP X x Y y

P y .

Conditional probability function Y given X ,

/( )

i i

P x yP Y y X x

P x .

,

/( )

P X a Y bP X a Y b

P Y b

3) Conditional density function of X given Y, ( , )

( / )( )

f x yf x y

f y .

Conditional density function of Y given X, ( , )

( / )( )

f x yf y x

f x .

4) If X and Y are independent random variables then



( , ) ( ). ( )f x y f x f y (for continuous random variable)

, .P X x Y y P X x P Y y (for discrete random variable)

5) Joint probability density function , ( , )

d b

c a

P a X b c Y d f x y dxdy .

0 0

, ( , )

b a

P X a Y b f x y dxdy

6) Marginal density function of X, ( ) ( ) ( , )X

f x f x f x y dy

Marginal density function of Y, ( ) ( ) ( , )Y

f y f y f x y dx

7) ( 1) 1 ( 1)P X Y P X Y

8) Correlation co – efficient (Discrete): ( , )

( , )X Y

Cov X Yx y

1( , )Cov X Y XY XY

n , 2 21

XX X

n , 2 21

YY Y

n

9) Correlation co – efficient (Continuous): ( , )

( , )X Y

Cov X Yx y

( , ) ,Cov X Y E X Y E X E Y , ( )X

Var X , ( )Y

Var Y

10) If X and Y are uncorrelated random variables, then ( , ) 0Cov X Y .

11) ( )E X xf x dx

, ( )E Y yf y dy

, , ( , )E X Y xyf x y dxdy

.

12) Regression for Discrete random variable:

Regression line X on Y is xyx x b y y ,

2xy

x x y yb

y y

Regression line Y on X is yxy y b x x ,

2yx

x x y yb

x x

Correlation through the regression, .XY YX

b b Note: ( , ) ( , )x y r x y



13) Regression for Continuous random variable:

Regression line X on Y is ( ) ( )xy

x E x b y E y , x

xy

y

b r

Regression line Y on X is ( ) ( )yx

y E y b x E x , y

yx

x

b r

Regression curve X on Y is / / x E x y x f x y dx

Regression curve Y on X is / / y E y x y f y x dy

14) Transformation Random Variables:

( ) ( )Y X

dxf y f x

dy (One dimensional random variable)

( , ) ( , )UV XY

x x

u vf u v f x y

y y

u v

(Two dimensional random variable)

15) Central limit theorem (Liapounoff’s form)

If X1, X2, …Xn be a sequence of independent R.Vs with E[Xi] = µi and Var(Xi) = σi2, i

= 1,2,…n and if Sn = X1 + X2 + … + Xn then under certain general conditions, Sn

follows a normal distribution with mean 1

n

i

i

and variance 2 2

1

n

i

i

as

n .

16) Central limit theorem (Lindberg – Levy’s form)

If X1, X2, …Xn be a sequence of independent identically distributed R.Vs with E[Xi]

= µi and Var(Xi) = σi2, i = 1,2,…n and if Sn = X1 + X2 + … + Xn then under certain

general conditions, Sn follows a normal distribution with mean n and variance

2n as n .



Note: nS n

zn

( for n variables),

Xz

n

( for single variables)

UNIT-III (MARKOV PROCESSES AND MARKOV CHAINS)

1) Random Process:

A random process is a collection of random variables {X(s,t)} that are

functions of a real variable, namely time ‘t’ where s Є S and t Є T.

2) Classification of Random Processes:

We can classify the random process according to the characteristics of time t

and the random variable X. We shall consider only four cases based on t and X

having values in the ranges -∞< t <∞ and -∞ < x < ∞.

Continuous random process

Continuous random sequence

Discrete random process

Discrete random sequence

Continuous random process:

If X and t are continuous, then we call X(t) , a Continuous Random Process.

Example: If X(t) represents the maximum temperature at a place in the

interval (0,t), {X(t)} is a Continuous Random Process.

Continuous Random Sequence:

A random process for which X is continuous but time takes only discrete values is

called a Continuous Random Sequence.

Example: If Xn represents the temperature at the end of the nth hour of a day, then

{Xn, 1≤n≤24} is a Continuous Random Sequence.

Discrete Random Process:

If X assumes only discrete values and t is continuous, then we call such random

process {X(t)} as Discrete Random Process.

Example: If X(t) represents the number of telephone calls received in the interval

(0,t) the {X(t)} is a discrete random process since S = {0,1,2,3, . . . }

Discrete Random Sequence:

A random process in which both the random variable and time are discrete is called

Discrete Random Sequence.



Example: If Xn represents the outcome of the nth toss of a fair die, the {Xn : n≥1} is a

discrete random sequence. Since T = {1,2,3, . . . } and S = {1,2,3,4,5,6}

3) Condition for Stationary Process: ( ) ConstantE X t , ( ) constantVar X t .

If the process is not stationary then it is called evolutionary.

4) Wide Sense Stationary (or) Weak Sense Stationary (or) Covariance Stationary: A random process is said to be WSS or Covariance Stationary if it satisfies the

following conditions.

i) The mean of the process is constant (i.e) ( ) constantE X t .

ii) Auto correlation function depends only on (i.e)

( ) ( ). ( )XX

R E X t X t

5) Time average:

The time average of a random process ( )X t is defined as1

( ) 2

T

T

T

X X t dtT

.

If the interval is 0,T , then the time average is 0

1( )

T

TX X t dt

T .

6) Ergodic Process:

A random process ( )X t is called ergodic if all its ensemble averages are

interchangeable with the corresponding time averageT

X .

7) Mean ergodic:

Let ( )X t be a random process with mean ( )E X t and time averageT

X ,

then ( )X t is said to be mean ergodic if T

X as T (i.e)

( )T

TE X t Lt X

.

Note: var 0T

TLt X

(by mean ergodic theorem)

8) Correlation ergodic process:

The stationary process ( )X t is said to be correlation ergodic if the process

( )Y t is mean ergodic where ( ) ( ) ( )Y t X t X t . (i.e) ( )T

TE Y t Lt Y

.

Where T

Y is the time average of ( )Y t .

9) Auto covariance function:

( ) ( ) ( ) ( )XX XX

C R E X t E X t

10) Mean and variance of time average:

Mean: 0

1( )

T

TE X E X t dt

T

Variance: 2

2

1( ) ( )

2

T

T XX XX

T

Var X R C dT



11) Markov process: A random process in which the future value depends only on the present value and not on the past values, is called a markov process. It is symbolically

represented by 1 1 1 1 0 0

( ) / ( ) , ( ) ... ( )n n n n n n

P X t x X t x X t x X t x

1 1( ) / ( )

n n n nP X t x X t x

Where 0 1 2 1

...n n

t t t t t

12) Markov Chain:

If for all n , 1 1 2 2 0 0

/ , , ...n n n n n n

P X a X a X a X a

1 1/

n n n nP X a X a

then the process n

X , 0,1, 2, ...n is called the

markov chain. Where 0 1 2, , , ... , ...

na a a a are called the states of the markov chain.

13) Transition Probability Matrix (tpm): When the Markov Chain is homogenous, the one step transition probability is denoted by Pij. The matrix P = {Pij} is called transition probability matrix.

14) Chapman – Kolmogorov theorem: If ‘P’ is the tpm of a homogeneous Markov chain, then the n – step tpm P(n) is

equal to Pn. (i.e) ( )n

n

ij ijP P .

15) Markov Chain property: If 1 2 3, , , then P and

1 2 31 .

16) Poisson process: If ( )X t represents the number of occurrences of a certain event in (0, )t ,then

the discrete random process ( )X t is called the Poisson process, provided the

following postulates are satisfied.

(i) 1 occurrence in ( , )P t t t t O t

(ii) 0 occurrence in ( , ) 1P t t t t O t

(iii) 2 or more occurrences in ( , )P t t t O t

(iv) ( )X t is independent of the number of occurrences of the event in any

interval.

17) Probability law of Poisson process:

( ) , 0,1, 2, ...!

xte t

P X t x xx

Mean ( )E X t t , 2 2 2( )E X t t t , ( )Var X t t .

UNIT-IV (CORRELATION AND SPECTRAL DENSITY)



XXR - Auto correlation function

XXS - Power spectral density (or) Spectral density

XYR - Cross correlation function

XYS - Cross power spectral density

1) Auto correlation to Power spectral density (spectral density):

i

XX XXS R e d

2) Power spectral density to Auto correlation:

1

2

i

XX XXR S e d

3) Condition for ( )X t and ( )X t are uncorrelated random process is

( ) ( ) ( ) ( ) 0XX XX

C R E X t E X t

4) Cross power spectrum to Cross correlation:

1

2

i

XY XYR S e d

5) General formula:

i) 2 2cos cos sin

ax

ax ee bx dx a bx b bx

a b

ii) 2 2sin sin cos

ax

ax ee bx dx a bx b bx

a b

iii) 2 2

2

2 4

a ax ax x

iv) sin2

i ie e

i

v) cos2

i ie e



UNIT-V (LINEAR SYSTEMS WITH RANDOM INPUTS)

1) Linear system:

f is called a linear system if it satisfies

1 1 2 2 1 1 2 2( ) ( ) ( ) ( )f a X t a X t a f X t a f X t

2) Time – invariant system:

Let ( ) ( )Y t f X t . If ( ) ( )Y t h f X t h then f is called a time –

invariant system.

3) Relation between input ( )X t and output ( )Y t :

( ) ( ) ( ) Y t h u X t u du

Where ( )h u system weighting function.

4) Relation between power spectrum of ( )X t and output ( )Y t :

2( ) ( ) ( )

YY XXS S H

If ( )H is not given use the following formula ( ) ( ) j t

H e h t dt

5) Contour integral:

2 2

imxm ae

ea x a

(One of the result)

6) 1

2 2

1

2

ae

Fa a

(from the Fourier transform)


Engineering Mathematics Material 2012


SUBJECT NAME : Statistics

MATERIAL NAME : Formula & Problems

MATERIAL CODE : JM08AM1003


UNIT – I (Testing of Hypothesis)

1. Write the application of ‘F’ - test and – test and ‘t’ – test.

Application of – test

i) To test the goodness of fit.

ii) To test the “independence of attributes”.

iii) To test the significance of discrepancy between experimental values and the

theoretical values.

Application of ‘F’ – test i) To test whether there is any significant difference between two estimates of

population variance.

ii) To test if the two samples have come from the same population.

Application of ‘t’ – test i) Test of Hypothesis about the population mean.

ii) Test of Hypothesis about the difference between two means.

iii) Test of Hypothesis about the difference between two means with dependent

samples.

iv) Test of Hypothesis about the observed sample correlation coefficient and sample

regression coefficient.

2. Define Type – I and Type – II errors and critical region.

Type – I Error

The hypothesis is true but our test rejects it.

Type – II Error

The hypothesis is false but our test accepts it.

Critical region

A region in the sample space which amounts to rejection of H0 is termed as

critical region or rejection.



3. Define F – variate.

Greater VarianceF

Smaller Variance

2

2

1

11

x xS

n

and

2

2

2

21

y yS

n

Here 2

1S and 2

2S are Variance

and 1S and 2

S are S.D.

4. Write down any two properties of chi – Square distribution.

i) The mean and variance of the chi – Square distribution are n and 2n respectively.

ii) As n → , Chi – Square distribution approaches a normal distribution.

iii) The sum of independent Chi – Square variates is also a Chi – Square variate.

5. What are the assumptions for Student’s ‘t’ test?

i) The parent population from which the sample drawn is normal.

ii) The sample observations are independent.

iii) The population standard deviation is unknown.

UNIT – I (Testing of Hypothesis) There are two types of samples in Testing of Hypothesis

i) Small Sample (n <30)

ii) Large Sample (n >30)

Small Sample: (n <30)

Type 1 Student’s ‘t’ test

Single Mean (S.D given directly)

Single Mean (S.D is not given directly)

Difference of Means

Type 2 F – test

Type 3 – test

Goodness of fit



Independence of attributes

Large Sample: (n >30)

Type 1 Single Mean

Type 2 Single Proportion

Type 3 Difference of Mean

Type 4 Difference of Proportion

Small Sample: (n <30)

Student’s ‘t’ test:

Single Mean (S.D given directly)

√

Where Sample Mean

Population

n Sample size

S.D Standard deviation

Degree of freedom = n – 1

Single Mean (S.D is not given directly)

√

Where ∑( )

Degree of freedom = n – 1

Difference of Means

√

Where



[∑( )

∑( )

]

(or)

Where 2 2

2 1 1 2 2

1 22

n s n ss

n n

Degrees of freedom 1 22n n

F - test: Greater Variance

FSmaller Variance

2

2

1

11

x xS

n

and

2

2

2

21

y yS

n

Here 2

1S and 2

2S are Variance and

1S and 2

S are S.D. (Capital S mention Population S.D and Small s mention Sample S.D)

Chi – Square test :

Goodness of fit:

2

2O E

E

Where O – Observed frequency

E – Expected frequency, Degree of freedom 1n

Independence of attributes:

a b a + b

c d c + d

a + c b + d N

The expected frequencies are given by ( )( )

( )( )

a + b

( )( )

( )( )

c + d

a + c b + d N

Degree of freedom 1 1r s

Where r – number of rows



S – number of columns

Large Sample: (n >30)

Single Mean (or) x x

zs

n n

Difference of Mean 1 2 1 2

2 2 2 2

1 2 1 2

1 2 1 2

(or) x x x x

zs s

n n n n

Single Proportion

p Pz

PQ

n

where p – Sample proportion, P – Population proportion, Q = 1 – P

Difference of Proportion

1 2

1 2

1 1

p pz

pqn n

where 1 1 2 2

1 2

n p n pp

n n

and 1q p


Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 1

SUBJECT NAME : Discrete Mathematics



MATERIAL CODE : JM08ADM009 (Scan the above Q.R code for the direct download of this material)


Unit – I (Logic and Proofs) 1) Truth Table:

Conjunction Disjunction Conditional Biconditional

p

q

p q

p

q

p q

p

q

p q

p

q

p q

T T T T T T T T T T T T

T F F T F T T F F T F F

F T F F T T F T T F T F

F F F F F F F F T F F T

Negation

p p

T F

F T

2) Tautology and Contradiction:

A Compound proposition 1 2, , ...

nP P P P where

1 2, , ...

nP P P variables are called

tautology if it is true for every truth assignment for 1 2, , ...

nP P P .

P is called a Contradiction if it is false for every truth assignment for 1 2, , ...

nP P P .

If a proposition is neither a tautology nor a Contradiction is called contingency.

3) Laws of algebra of proposition:

Name of Law Primal form Dual form



Idempotent law p p p p p p

Identity law p F p p T p

Dominant law p T T p F F

Complement

law p p T p p F

Commutative

law p q q p p q q p

Associative law p q r p q r p q r p q r

Distributive law p q r p q p r p q r p q p r

Absorption law p p q p p p q p

Demorgan’s law p q p q p q q p

Double

Negation law

p p

4) Equivalence involving Conditionals:

Sl.No. Propositions

1. p q p q

2. p q q p

3. p q p q

4. p q p r p q r

5. p r q r p q r

5) Equivalence involving Biconditionals:

Sl.No. Propositions

1. p q p q q p

2. p q p q



3. p q p q p q

4. p q p q

6) Tautological Implication:

A B if and only if A B is tautology. (i.e) To prove A B , it enough to prove

A B is tautology.

7) The Theory of Inferences:

The analysis of the validity of the formula from the given set of premises by using

derivation is called “theory of inferences”

8) Rules for inferences theory:

Rule P:

A given premise may be introduced at any stage in the derivation.

Rule T:

A formula S may be introduced in a derivation if S is tautologically implied by one or

more of the preceding formulae in the derivation.

Rule CP:

If we can drive S from R and a set of given premises, then we can derive R S from

the set of premises alone. In such a case R is taken as an additional premise (assumed

premise). Rule CP is also called the deduction theorem.

9) Indirect Method of Derivation:

Whenever the assumed premise is used in the derivation, then the method of derivation

is called indirect method of derivation.

10) Table of Logical Implications:

Name of Law Primal form

Simplification

p q p

p q q

Addition

p p q

q p q

Disjunctive Syllogism

p p q q

q p q p

Modus Ponens p p q q

Modus Tollens p q q p

Hypothetical Syllogism p q q r p r



p q q p

Unit – II (Combinatorics) 1) Principle of Mathematical Induction:

Let ( )P n be a statement or proposition involving for all positive integers n.

Step 1: (1)P is true.

Step2: Assume that ( )P k is true.

Step3: We have to prove ( 1)P k is true.

2) Principle of Strong induction.

Let ( )P n be a statement or proposition involving for all positive integers n.

Step 1: (1)P is true.

Step2: Assume that ( )P n is true for all integers 1 n k .

Step3: We have to prove ( 1)P k is true.

3) The Pigeonhole Principle:

If n pigeons are assigned to m pigeonholes and m n , than at least one pigeonhole

contains two or more pigeons.

4) The Extended Pigeonhole Principle:

If n pigeons are assigned to m pigeonholes than one pigeonhole must contains at least

11

n

m

pigeons.

5) Recurrence relation:

An equation that expressesn

a , the general term of the sequence na in terms of one or

more of the previous terms of the sequence, namely0 1 1, , ...

na a a

, for all integers n is

called a recurrence relation for na or a difference equation.

6) Working rule for solving homogeneous recurrence relation:

Step 1: The given recurrence relation of the form

0 1 1( ) ( ) ... ( ) 0

n n k n kC n a C n a C n a

Step 2: Write the characteristic equation of the recurrence relation

0 1 1... 0

n k k nn kC r C r C r

Step 3: Find all the roots of the characteristic equation namely1 2, , ...

kr r r .

Step 4:

Case (i): If all the roots are distinct then the general solution is

1 1 2 2...

n n n

n k ka b r b r b r

Case (ii): If all the roots are equal then the general solution is

2

1 2 3...

n

na b nb n b r



Unit – III (Graph Theory) 1) Graph:

A graph G=(V,E) consists of two sets 1 2, , ...V v v v , called the set of vertices and

1 2, , ...E e e e

, called the set of edges of G.

2) Simple graph: A graph is said to be simple graph if it has no loops and parallel edges. Otherwise it is

multi graph.

3) Regular graph: If every vertex of a simple graph has the same degree, then the graph is called a regular

graph. If every vertex in a regular graph has degree n, then the graph is called n-regular. 4) Complete graph:

A simple graph in which each pair of distinct vertices is joined by an edge is called a

complete graph. The complete graph on “n” vertices is denoted byn

K .

5) Pendent vertex and Pendent edge:

A vertex with degree one is called a pendent vertex and the only edge which is incident

with a pendent vertex is called the pendent edge.

6) Matrix representation of a graph:

There are two ways of representing a graph by a matrix namely adjacent matrix and

incidence matrix as follows:

Adjacency matrices:

Let G be a graph with n vertices, then the adjacency matrix, G ijA A defined

by 1, if , are adjacent

0, otherwise

i j

ij

u vA

.

Incidence matrix:

Let G be a graph with n vertices, then the incidence matrix of G is an n x e matrix

G ijB B defined by

1, if edge is incident on the vertex

0, otherwise

th th

ij

j iB

7) Bipartite graph:

A graph G=(V,E) is called a bipartite graph if its vertex set V can be partitioned into two

subsets 1

V and 2

V such that each edge of G connects and vertex of 1

V to a vertex of 2

V .

In other words, no edge joining two vertices, in 1

V or two vertices in 2

V .

8) Isomorphism of a graph:

The simple graphs 1 1 1,G V E and 2 2 2

,G V E are isomorphic if there is a one to

one and onto function f from 1

V to 2

V with the property that a and b are adjacent in

1G if and only if ( )f a and ( )f b are adjacent in

2G , for all a and b in

1V .



9) Complementary and Self complementary graph:

Let G be a graph. The complement G of G is defined by any two vertices are adjacent

in G if and only if they are not adjacent inG .

G is said to be a self complementary graph if G is isomorphic toG .

10) Connected graph:

A graph G is said to be connected if there is at least one path between every pair of

vertices in G. Otherwise G is disconnected. A disconnected graph consists of two or

more connected sub graphs and each of them is called a component. It is denoted by

( )G .

11) Strongly Connected and Weakly Connected graph:

12) Cut edge:

A cut edge of a graph G is an edge “e” such that ( ) ( )G e G . (i.e) If G is

connected and “e” is a cut edge of G, then G e is disconnected.

13) Cut vertex:

A cut vertex of a graph G is a vertex “v” such ( ) ( )G v G . (i.e) If G is connected

and “v” is a cut vertex of G, then G v is disconnected.

14) Define vertex connectivity.

The connectivity ( )G of G is the minimum “k” for which G has a k-vertex cut. If G is

either trivial or disconnected then ( ) 0G .

15) Define edge connectivity.

The edge connectivity ( )G of G is the minimum “k” for which G has a k-edge cut. If G

is either trivial or disconnected then ( ) 0G

16) Define Eulerian graph.

A path of graph G is called an Eulerian path, if it includes each edge of G exactly once. A

circuit of a graph G is called an Eulerian circuit, if it includes each edge of G exactly one.

A graph containing an Eulerian circuit is called an Eulerian graph.

17) Define Hamiltonian graph.

A simple path in a graph G that passes through every vertex exactly once is called a

Hamilton path. A circuit in a graph G that passes through every vertex exactly once is

called a Hamilton circuit. A graph containing a Hamiltonian circuit.

Unit – IV (Algebraic Structures) 1) Semi group:

If G is a non-empty set and * be a binary operation on G, then the algebraic system

,*G is called a semi group, if G is closed under * and * is associative.

Example: If Z is the set of positive even numbers, then ,Z and ,Z are semi

groups.



2) Monoid:

If a semi group ,*G has an identity element with respect to the operation *, then

,*G is called a monoid. It is denoted by ,*,G e .

Example: If N is the set of natural numbers, then ,N and ,N are monoids with

the identity elements 0 and 1 respectively. ,Z and ,Z are semi groups with

out monoids, where Z is the set of all positive even numbers 3) Sub semi groups:

If ,*G is a semi group and H G is called under the operation *, then ,*H is

called a sub semi group of ,*G .

Example: If the set E of all even non-negative integers, the ,E is a sub semi group

of the semi group ,N , where N is the set of natural numbers. 4) Semi group homomorphism:

If ,*G and ,G are two semi groups, then a mapping :f G G is called a semi

group homomorphism, if for any ,a b G , ( * ) ( ) ( )f a b f a f b . A homomorphism f

is called isomorphism if f is 1-1 and onto. 5) Group:

If G is a non-empty set and * is a binary operation of G , then the algebraic system

,*G is called a group if the following conditions are satisfied.

(i) Closure property

(ii) Associative property

(iii) Existence of identity element

(iv) Existence of inverse element

Example: ,Z is a group and ,Z is not a group.

6) Abelian group: A group ,*G , in which the binary operation * is commutative, is called a

commutative group or abelian group.

Example: The set of rational numbers excluding zero is an abelian group under the

multiplication. 7) Coset:

If H is a subgroup of a group G under the operation *, then the set aH, where a G ,

define by * /aH a h h H is called the left coset of H in G generated by the

element a G . Similarly the set Ha is called the right coset of H in G generated by the

element a G .



Example: 1, 1, ,G i i be a group under multiplication and 1, 1H is a

subgroup of G. The right cosets are 1 1, 1H , 1 1,1H , ,iH i i and

,iH i i . 8) Lagrange’s theorem:

The order of each subgroups of a finite group is a divisor of a order of a group. 9) Cyclic group:

A group ,*G is said to be cyclic, if and element a G such that every element of

G generated by a. (i.e) 21, , , ...

nG a a a a e .

Example: 1, 1, ,G i i is a cyclic group under the multiplication. The generator is

i , because 4 2 31, 1, , i i i i i .

10) Normal subgroup: A subgroup H of the group G is said to be normal subgroup under the operation *, if for

any a G , aH Ha . 11) Kernel of a homomorphism:

If f is a group homomorphism from ,*G and ,G , then the set of element of G ,

which are mapped into e , the identity element of G , is called the kernel of the

homomorphism f and denoted by ker f . 12) Fundamental theorem of homomorphism:

If f is a homomorphism of G on to G with kernel K, then /G K is isomorphic to G . 13) Cayley’s theorem:

Every finite group of order n is isomorphic to a permutation group of degree n . 14) Ring:

An algebraic system , ,S is called a ring if the binary operations and on S

satisfy the following properties.

(i) ,S is an abelian group

(ii) ,S is a semi group

(iii) The operation is distributive over .

Example: The set of all integers Z , and the set of all rational numbers R are rings

under the usual addition and usual multiplication. 15) Commutative ring:

16) Integral domain: A commutative ring without zero divisor is called Integral domain.

Example: (i) , ,R is an integral domain, since ,a b R such that

0, 0a b then 0ab . (ii) 10 10 10, ,Z is not an integral domain,



because 10

2,3 Z and 10

2 5 0 . Therefore 2 and 5 are zero divisors.

17) Field:

A commutative reing , ,S which has more than one element such that every non-

zero element of S has a multiplicative inverse in S is called a field.

Example: The ring of rational numbers , ,Q is a field since it is a commutative ring

and each non-zero element is inversible.

Unit – V (Lattices and Boolean algebra) 1) Partially ordered set (Poset):

A relation R on a set A is called a partial order relation, if R is reflexive, antisymmetric

and transitive. The set A together with partial order relation R is called partially ordered

set or poset.

Example: The greater than or equal to relation is a partial ordering on the set of

integers Z . 2) Lattice:

A lattice is a partially ordered set ,L in which every pair of elements ,a b L has a

glb and lub. 3) Sub-lattice:

4) General formula:

i) glb , *a b a b a b

ii) l b ,u a b a b a b

iii) *

& *

a b a

a b b

&

a b a

a b b

iv) If *

a b a b a

a b b

5) Properties:

Name of Law Primal form Dual form

Idempotent law *a a a a a a

Commutative

law * *a b b a a b b a

Associative law * * * *a b c a b c a b c a b c



Distributive law * * *p q r p q p r * *p q r p q p r

Absorption law *a a b a *a a b a

Complement * 0a a 1a a

Demorgan’s law *a b a b *a b a b

Double

Negation law

p p

6) Complemented Lattices:

A Lattice ,*,L is said to be complemented if for any a L , there exist a L ,

such that * 0a a and 1a a .

7) Demorgan’s laws:

Let ,*,L be the complemented lattice, then *a b a b and

*a b a b .

8) Complete Lattice: A lattice ,*,L is complete if for all non-empty subsets of L, there exists a glb and

lub.

9) Lattice Homomorphism: Let ,*,L and , ,S be two lattices. A mapping :g L S is called lattices

homomorphism if ( * ) ( ) ( )g a b g a g b and ( ) ( ) ( )g a b g a g b . 10) Modular Lattice:

A lattice ,*,L is said to be modular if for any , ,a b c L

i) * *a c a b c a b c

ii) * *a c a b c a b c

11) Chain in Lattice: Let ,L be a Chain if

i) or a b a c and ii) and a b a c

12) Condition for the algebraic lattice: A lattice ,*,L is said to be algebraic if it satisfies Commutative Law, Associative

Law, Absorption Law and Existence of Idempotent element. 13) Isotone property:



Let ,*,L be a lattice. The binary operations * and are said to possess isotone

property if * *

b c a b a c

a b a c

.

14) Boolean Algebra:

A Boolean algebra is a lattice which is both complemented and distributive. It is denoted

by ,*,B .


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