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[BABU SRINIVASAN]

MA131 MATHEMATICS I
QUESTION BANK
Question Bank
MATRICES:
PART –A
1. Find the eigen values of 2A+I given that ⎟ ⎟
⎞
⎜ ⎜
3 2
.
⎝ ⎠
⎛
=
4 1
A
2. Find the eigen values and eigen vectors of ⎟
⎟⎠
⎞
⎜ ⎜
⎝
⎛
=
0
0
a
a
A .
3. Find the sum of the eigen values of 2A if
⎟ ⎟ ⎟
⎠
⎞
⎜ ⎜ ⎜
⎝
⎛
−
− −
−
=
2 4 3
6 7 4
8 6 2
A .
4. Verify Cayley- Hamilton theorem for ⎟ ⎟ .
⎠
⎞
⎜ ⎜⎝
⎛
=
1 3
5 3
A
5. Find the matrix of the quadratic form x2 + y2 + z2 + 2xz + 4 2yz .
PART - B
6. (i) Using Cayley-Hamilton theorem, find the inverse of A where
(8 Marks)
⎟ ⎟ ⎟
⎠
⎞
⎜ ⎜ ⎜
⎝
⎛
− −
= −
2 4 4
1 3 3
1 1 3
A
(ii) Using diagonalization, find A6 given that . (8 Marks) ⎟
⎟⎠
⎞
⎜ ⎜⎝
⎛
=
1 2
5 4
A
7. Find an orthogonal transformation which reduces the quadratic form
2xy + 2yz + 2xz to a canonical form. Also find its nature, index, signature and
rank. (16 Marks)
7. Reduce the quadratic form xz to a canonical
form by an orthogonal transformation. Also find its nature, index, signature and
rank. (16 Marks)
2x2 + y2 + z2 + 2xy − 4yz − 2
8. (i) Using Cayley-Hamilton theorem, find An given that ⎟ ⎟ . Hence
find A3. (8 Marks)
⎠
⎞
⎜ ⎜⎝
⎛
=
2 3
1 4
A
(ii) Diagonalize A by an orthogonal transformation where .
⎟ ⎟ ⎟
⎠
⎞
⎜ ⎜ ⎜
⎝
⎛
=
4 0 2
0 6 0
2 0 4
A
(8 Marks)
9. Find the characteristic equation of the matrix A and hence find the matrix
respresented by A8 − 5A7 + 7A6 − 3A5 + A4 − 5A3 + 8A2 − 2A + I
⎟ ⎟ ⎟
⎠
⎞
where
. (16 Marks)
⎜ ⎜ ⎜
⎝
⎛
=
1 1 2
0 1 0
2 1 1
A
ANALYTICAL GEOMETRY:
PART-A
1. A, B, C, D are the points (-3, 2, k), (4, 1, 6), (-1, -2, -3) and
(13, -4, -1). Find the
value of k if AB is parallel to CD.
2. Find the angle between the planes 2x − y + z + 7 = 0 and
x + y + 2z −11 = 0 .
3. Find the value of k if the lines
k
z x y z
k
x y
−
=
+
=
−
+ +
=
−
=
−
−
3
1
4
& 1
5
3 1
1
1
are perpendicular.
4. Find the equation of the sphere having the points (2,-3,4) and
(-1,5,7) as the ends
of a diameter.
5. Find the equation of the tangent plane at the point (1,-1,2) to the sphere
x2 + y2 + z2 − 2x + 4y + 6z −12 = 0 .
PART-B
6. (i) Prove that the four points A(2,5,3), B(7,9,1), C(3,-6,2), D(13,2,-2) are
coplanar. (8 Marks)
(ii) Find the equation of the plane that contains the parallel lines
3
4
2
2
1
& 3
3
3
2
2
1
1 +
=
+
=
− −
=
−
=
x − y z x y z
. (8 Marks)
7. (i) Find the length of the shortest distance line between the lines
; 2 3 5 6 0 3 2 3
3 4
1
2
2 = + − − = = − − +
+
=
− x y z x y z x y z
. (8 Marks)
(ii) Find the co-ordinates of the foot, the length and the equations of the
perpendicular from the point (-1,3,9) to the line
1
31
8
8
5
13 −
=
−
+
=
x − y z
.
(8 Marks)
2
8. (i) Show that plane 2x − 2y + z +12 = 0 touches the sphere
x2 + y2 + z2 − 2x − 4y + 2z = 3 and find also the point of contact. (8 Marks)
(ii) Find the equation of the sphere having the circle
x2 + y2 + z2 +10y − 4z − 8 = 0; x + y + z = 3as a great circle. (8 Marks)
9. (i) Find the equation of the sphere that passes through the circle
1 0 and cuts
orthogonally the sphere 2 0. (8 Marks)
x2 + y2 + z2 + 2x + 3y + z − 2 = 0; 2x − y − 3z − =
x2 + y2 + z2 − 3x + y − =
(ii) Find the centre and radius of the circle given by
x2 + y2 + z2 + 2x − 2y − 4z −19 = 0; x + 2y + 2z + 7 = 0 . (8 Marks)
(i) Find the length and equations of the shortest distance line
between the lines
; 3 2 5 6 0 2 3 3
2 1
2
3
1 = + − − = = − + −
−
=
+ x y z x y z x y z
. (8 Marks)
(ii) Find the equation of the sphere passing through the circle given by
and
and the point (1,-2,3). (8 Marks)
x2 + y2 + z2 + 3x + y + 4z − 3 = 0 x2 + y2 + z2 + 2x + 3y + 6 = 0
GEOMETRICAL APPLICATIONS OF DIFFERENTIAL CALCULUS
PART-A
1. Find the curvature of the curve given by y = c tan x at x = 0.
2. Find the radius of curvature of the curve xy = c2.
3. Find the radius of curvature at the point (p,r) of the ellipse
2 2
2
2 2 2
1 1 1
a b
r
p a b
= + − .
4. Find the envelope of the family of lines
m
y = mx + a , where m is a parameter.
5. Show that the radius of curvature of a circle is its radius.
PART-B
6. (i) Find the measure of curvature of the curve + =1
b
y
a
x
at any point (x,y)
on it. (8 Marks)
(ii) Find the centre of curvature at 2
θ =π on the curve
x = 2cost + cos2t, y = 2sint + sin2t . (8 Marks)
3
7. (i) Find the equation of the circle of curvature of the parabola x at the
point (3,6).
y2 =12
(ii) Find the radius of curvature of the curve r = a(1+ cosθ ) at the point
2
θ =π . (8 Marks)
8. (i) If ϕ be the angle which the radius vector of the curve r = f (θ
) makes with
the tangent, then prove that ⎟⎠
⎞
⎜⎝
= ⎛ +
θ
ϕ
ϕ
ρ d
r sin 1 d , where ρ is the radius of
curvature. Apply this result to show that ρ = a/2 for the circle r = acosθ .
(8 Marks)
(ii) Find the radius of curvature at the origin for the cycloid
x = a(θ + sinθ ); y = a(1− cosθ ) . (8 Marks)
9. (i) Find the envelope of the straight lines
ysint + xcost = a + acost logtan(t / 2) . (8 Marks)
(ii) Find the radius of curvature at the point θ on
x = 3acosθ − acos3θ , y = 3asinθ − asin3θ. (8 Marks)
10. (i) Find the evolute of the astroid x = acos3θ , y = asin3θ . (8 Marks)
(ii) Show that the envelope of a circle whose centre lies on the parabola
and which passes through its vertex is the cissoid
. (8 Marks)
y2 = 4ax
y2 (2a + x) + x3 = 0
FUNCTIONS OF SEVERAL VARIABLES:
PART-A
1. Evaluate
y
z
x
z
∂
∂
∂
∂
& if x + y + z = log z .
2. If u = log(x + y + z) & x = e−t , y = sin t, z = cost , then find
dt
du
.
3. Find
dx
du
, when u = sin(x2 + y2 ) & x2 + 4y2 = 9 .
4. Write down the Maclaurin's series for sin(x + y)
5. If
y
v x
x
u y
2 2
= , = , then find
( , )
( , )
u v
x y
∂
∂
.
4
PART-B
6. (i) Expand x2 y + 3y − 2 in powers of (x-1) and (y+2) using
Taylor's expansion.
(8 Marks)
(ii) Find the maximum and minimum values of x2 − xy + y2 − 2x + y .
(8 Marks)
7. (i) Find the volume of the greatest rectangular parallelepiped that
can be inscribed
in the ellipsoid 1 2
2
2
2
2
2
+ + =
c
z
b
y
a
x
. (8 Marks)
(ii) Prove that (1 ) log(1 )
0
e dx a
x
e ax
x
+ = − ∫
∞
−
−
where a>-1 using differentiation
under the integral sign. (8 Marks)
8. (i) If z = f(u,v) where v xy, then show that
). (8 Marks)
u = x2 − y2 , = 2
4( 2 2 )( vv
zxx + zyy = x + y zuu + z
(ii) By using the transformations u = x + y, v = x − y
0
, change the independent
variables x and y in the equation zxx − zyy = to u and v. (8 Marks)
9. (i) If x2 + y2 + z2 − 2xyz =1, show that
0
1 2 1 2 1 2
=
−
+
−
+
− z
dz
y
dy
x
dx
. (8 Marks)
(ii) Expand ex cos y in powers of x and y as far as the terms of third degree.
(8 Marks)
10. (i)A rectangular box, open at the top, is having a volume of 32c.c. Find the
dimensions of the box, that requires the least material for its construction.
(8 Marks)
(ii) Expand ex log(1 + y) in powers of x and y as far as the terms of
third degree.
(8 Marks)
DIFFERENTIAL EQUATIONS:
PART-A
1. Solve (D − 2)2 y = e2x .
2. Find the Particular Integral of (D + 2)2 y = e−x cos x .
3. Solve (D2 + 4) y = sin 2x .
5
4. Solve n x
dt
d x 4
4
4
= .
5. Find the Particular Integral of (D − 3)2 y = xe−2x .
PART-B
6. (i) Solve y e e3x sin x . (8 Marks)
dx
dy
dx
d y 4 4 2x
2
2
+ + = − +
(ii) Solve by the method of variation of parameters y x
dx
d y 4 4tan 2 2
2
+ = .
(8 Marks)
7. (i) Solve the simultaneous equations x t
dt
y t dy
dt
dx + = sin ; + = cos given that
x = 2, y = 0 when t = 0. (8 Marks)
(ii) Solve y x
dx
d y sec 2
2
+ = by the method of variation of parameters.(8 Marks)
8. Solve x y e t
dt
x y t dy
dt
dx + 2 − 3 = 5 ; − 3 + 2 = 2 2 . (16 Marks)
9. Solve the equation y x x
dx
x dy
dx
d y 2
2
2
+ (1 − cot ) − cot = sin by the method of
reduction of order. (16 Marks)
10. Solve 4sin(log ) 2
2
2 y x
dx
x dy
dx
x d y + + = . (16 Marks)
--End--
6
MA132 MATHEMATICS – II
QUESTION BANK
MULTIPLE INTEGRALS:
PART- A
1. Evaluate
1 1
a b dxdy
∫ ∫ xy .
2. Evaluate ∫ ∫ .
π θ
θ
0
sin
0
a
r dr d
3. Change the order of integration in∫ ∫
0 −
0
1
0
2
( , )
x
f x y dx dy.
4. Define Gamma function and Beta function.
5. Show thatβ (m,n) = β (n,m) .
PART-B
6. (i) Prove that
( )
( , ) ( ) ( )
m n
m n m n
Γ +
Γ Γ
β = .
(ii) Evaluate . ∫
∞
−
0
e x x7dx
7. (i) Find the volume bounded by the cylinder 2 4and the planes
+ z = 4 and z = 0 .
x2 + y =
y
(ii) Evaluate ∫∫ x y dx dy over the positive quadrant of the circle x2 + y2 =1.
8. (i) Change the order of integration dx dy
x y
I x
a a
y
∫ ∫ +
=
0
2 2 and then evaluate.
(ii) Evaluate I x y dx dy
a
a
a x
∫ ∫
−
−
= +
2 2
0
( 2 2 ) .
9. (i) Evaluate = ∫ .
/ 2
0
sin6 cos7
π
I θ θ dθ
(ii) Evaluate = ∫
/ 2
0
cot
π
I θ dθ .
10. (i) Evaluate and prove ∫
∞
= −
0
2 dx e I x π = ⎟⎠
⎞
⎜⎝
Γ⎛
2
1
.
(ii) Prove that Γ(n +1) = n!, when n is a positive integer.
VECTOR ANALYSIS:
PART – A
1. Find a unit normal to the surface x2 + y2 = z at (1,2,5) .
2. Find directional directive of x2 + y2 + z2 at (1,0,1) in the
direction 2i + 3k .
3. Prove thatcurl(gradφ ) = 0 .
4. Evaluate∇2 (logr) .
􀁇 􀁇
5. If∇φ = yz i + zx j + xy k ,findφ .
􀁇
PART – B
6. (i) Show that F xy z i x z j xz y k
􀁇 􀁇 􀁇 􀁇
= (6 + 3 ) + (3 2 − ) + (3 2 − ) is irrotational. Find
φ such that F = ∇φ
􀁇
.
(ii) Find the work done when a force F x y x i xy y j
􀁇 􀁇 􀁇
= ( 2 − 2 + ) − (2 + ) moves a
particle in the xy plane from (0,0)to (1,1) along the curve y2 = x .
7. (i) Prove that the area bounded by a simple closed curve C is given by
∫ −
C
(x dy y dx).
2
1
Hence find the area of the ellipse.
(ii) Find the area between y2 = 4x and x2 = 4y by using Green's theorem.
8. Verify Gauss's divergence theorem for F x i y j z k
􀁇 􀁇 􀁇 2 􀁇 taken over the cube
bounded by planes
= 2 + 2 +
x = 0, x =1, y = 0, y =1, z = 0and z =1.
9. Verify Green's theorem for ]where C is the boundary of the
common area between y = x2 and
[(xy y2 )dx x2dy
C
∫ + +
y = x.
10. Verify Stoke's theorem for a vector field F x y i xy j
􀁇 􀁇 􀁇
2 in the rectangular
region of the plane z = 0 bounded by the lines y b
= ( 2 − 2 ) +
x = 0, x = a, y = 0and = .
ANALYTIC FUNCTIONS:
PART – A
1. Find the invariant points of the transformation 3 5
1
w z
z
−
=
+
.
2. Prove that an analytic function with constant real part is constant.
3. Find the Bilinear transformation which maps z = 0,1, 2 into the points
w = 2,−1,3 respectively.
4. Test whether the function
z
f (z) = 1 is analytic or not.
5. Define: Conformal mapping.
PART-B
6. (i) Verify that 2 2
2 2
v x y x
x y
= − +
+
is harmonic and find u such that
w=u+iv is analytic. Express was a function of z .
(ii) Find the bi-linear transformation which maps 1,i, −1 in -plane to of the
-plane.
z 0,1,∞
w
7. (i) Under the transformation w=z2 , obtain the map in the w-plane
of the square
with vertices (0,0),(2,0),2,2)and (0,2) in z -plane.
(ii) Under the transformation
w 1
z
= find the image of the circle z + 1 =1and
z − 2i = 2 .
8. If f (z) is an analytic function of z , prove 2
2 2
( ) f (z) f (z)
y
f z
x
= ′
⎭ ⎬ ⎫
⎩ ⎨ ⎧
∂
∂
+
⎭ ⎬ ⎫
⎩ ⎨ ⎧
∂
∂
9. If f (z) = u + iv is an analytic function find f (z) and v if
x y
u x
cos2 cosh 2
sin 2
+
= .
10. If f (z) = u + iv is an analytic function find f (z) given that
h y x
u v x
cos 2 cos2
sin 2
−
+ = .
COMPLEX INTEGRATION:
PART – A
1. Evaluate dz
z
z
c ∫
+ 2
where C is z = 3.
2. Find the singular points of
z z
f z
sin
( ) = 1 .
3. Expand f (z) = ez in a Taylor's series about z = 0.
4. Find the residues at the isolated singularities of the functions
(z +1)(z − 2)
z .
5. Define essential singularity with an example.
PART – B
6. Evaluate the following integrals, using Cauchy's residue theorem
(i) ,
2 4
( 1)
∫ 2 + +
+
c z z
z dz where C is z +1+ i = 2.
(ii) ∫ c z +
dz
( 2 9)3
, where C is z − i = 3.
7. Find the Laurent's series of
(1 )
( ) 1
z z
f z
−
= valid in the region
(i) z +1 < 1, (ii)1 < z +1 < 2 and (iii) z +1 > 2
8. Use Cauchy's integral formula to evaluate
(i) dz
z z
z z
C ∫
− −
+
( 2)( 3)
sinπ 2 cosπ 2
, where C is the circle z = 4 .
(ii) dz
z z
z
C ∫
− −
−
3 4
7 1
2 , where c is the ellipse 1
4 1
2 2
x + y = .
9. (i) Evaluate
2
0 cos
π dθ
2 + θ ∫ using contour integration.
(ii) ) Evaluate ∫
∞
0 +
(1 x2 )2
dx
using contour integration
10. Evaluate
2
4 2
2
10 9
x x dx
x x
∞
−∞
− +
+ + ∫ using contour integration.
STATISTICS:
PART – A
1. Prove that the first moment about mean is always zero.
2. What is the difference between t distribution and Normal distribution?
3. What is correlation coefficient?
4. How is accuracy of regression equation ascertained?
5. Give two uses for χ 2 distribution.
PART-B
6. (i) Find the coefficient of correlation between X and Y using the
following data
: 16 19 23 26 30
: 5 10 15 20 25
Y
X
(ii) A study of prices of rice at Chennai and Madurai gave the following data:
Chennai Madurai
Mean 19.5 17.75
S.D. 1.75 2.5
Also the coefficient of correlation between the two is 0.8. Estimate the most
likely price of rice (i) at Chennai corresponding to the price of 18 at Madurai
and (ii) at Madurai corresponding to the price of 17 at Chennai.
7. (i) In a large city a, 20 % of a random sample of 900 school boys
had a slight
physical defect. In another large city B, 18.5 % of a random sample of 1600
school boys had the same defect. Is the difference between the proportions
significant?
(ii) A sample of 100 students is taken from a large population. The
mean height of
the students in this sample is 160cm. Can it be reasonably regarded that, in the
population, the mean height is 165 cm, and the S.D. is 10 cm?
8. (i) Samples of two types of electric bulbs were tested for length
of life and the
following data were obtained.
Size Mean S.D.
Sample I 8 1234 hours 36 hours
Sample II 7 1036 hours 40 hours
Is the difference in the means sufficient to warrant that type I bulbs
are superior to
type II bulbs?
(ii) Two samples of sizes nine and eight gave the sums of squares of deviations
from their respective means equal to 160 and 91 respectively. Can they be
regarded as drawn from the same normal population?
9. (i) Theory predicts that the proportion of beans in four groups A,
B,C, D should
be : 3: 3:1. In an experiment among 1600 beans, the numbers in the four
groups were 882, 313, 287 and 118. Does the experiment support the theory?
9
(ii) A sample of size 13 gave an estimated population variance of 3.0, while
another sample of size 15 gave an estimate of 2.5. Could both samples be from
populations with the same variance?
10. (i) A number of automobile accidents per week in a certain community are as
follows: 12, 8, 20, 2, 14, 10, 15, 6, 9, 4. Are these frequencies in agreement
with the belief that accident conditions were the same during this 10 week
period?
(ii) The mean of two random samples of size 9 and 7 are 196.42 and 198.82
respectively. The sums of the squares of the deviation from the mean are 26.94
and 18.73 respectively. Can the sample be considered to have been drawn
from the same normal population?
--End--
MA231 MATHEMATICS-III
QUESTION BANK
PARTIAL DIFFERENTIAL EQUATIONS
PART-A
1. Find the partial di(R)erential equation from (x¡a)2+(y¡b)2+z2 = r2
by eliminating the arbitrary
constants a and b.
2. Find the partial di(R)erential equation from z = x+y +f(xy) by
eliminating the arbitrary function
f.
3. Find the complete integral of pp + pq = 1.
4. Find the general solution of the Lagrange's equation px2 + qy2 = z2.
5. Solve (D + D0)(D + D0 + 1)z = 0.
PART-B
6.(i) Find the singular integral of the partial di(R)erential equation
z = px + qy + p2 + pq + q2.
(ii) Solve z2(p2 + q2) = x + y.
7.(i) Solve the equation x2(y ¡ z)p + y2(z ¡ x)q = z2(x ¡ y).
(ii) Form the partial di(R)erential equation by eliminating f from f(z
¡ xy; x2 + y2) = 0.
8.(i) Solve the equation (D2 + 4DD0 + D02)z = e2x¡y + 2x.
(ii) Solve the equation (D2 ¡ D02)z = sin(x + 2y) + ex¡y + 1:
9.(i) Solve the equation pq + p + q = 0.
(ii) Solve the Lagrange's equation (y + z)p + (z + x)q = x + y.
(iii) Solve (D3 ¡ D2D0 ¡ 8DD02 + 12D03)z = 0.
10.(i) Solve the equation pp + pq = x + y.
(ii) Formulate the partial di(R)erential equation by eliminating
arbitrary functions f and g from
z = f(x + ay) + g(x ¡ ay):
(iii) Solve p tan x + q tan y = tan z.
FOURIER SERIES
PART-A
1. State the Dirichlet's conditions for the existence of Fourier series of f(x).
2. Find the Fourier sine series of f(x) = x, 0 < x < ¼.
3. De¯ne Fourier series of f(x) in (c; c + 2l).
4. De¯ne the root mean square value of a function f(x) in (0; 2¼).
5. Find the Fourier coe±cient an, given that f(x) = x2 in (¡¼; ¼).
PART-B
6.(i) Find the Fourier series of f(x) = e¡x in (¡¼; ¼).
(ii) Find the half range cosine series of f(x) =
½
x; 0 < x < 1
2 ¡ x 1 < x < 2 .
1
7.(i) Obtain the Fourier series of the function given by f(x) =
8>><
>>:
1 +
2x
l
; ¡l · x · 0
1 ¡
2x
l
; 0 · x < l
.
(ii) Find the Fourier series of periodicity 2¼ for f(x) = x2, in ¡¼ <
x < ¼. Hence show that
1
14 +
1
24 +
1
34 + : : : = ¼4
90
.
8.(i) Find the Fourier sine series of f(x) = x(¼ ¡ x), 0 < x < ¼.
(ii) Compute the fundamental and ¯rst harmonics of the Fourier series
of f(x) given by the table.
x 0 ¼=3 2¼=3 ¼ 4¼=3 5¼=3 2¼
f(x) 1:0 1:4 1:9 1:7 1:5 1:2 1:0
9.(i) Express f(x) = (¼ ¡ x)2 as a Fourier series in 0 < x < 2¼.
(ii) Find the Fourier series of periodicity 2 for the function f(x) =
8<
:
k; ¡l · x · 0
x; 0 · x < l
.
10.(i) Find the half-range sine series of f(x) = l¡x in (0; l). Hence prove that
1
12 +
1
22 +
1
32 + : : : = ¼2
6
.
(ii) Find the constant term and the ¯rst two harmonics of the Fourier
cosine series of y = f(x) using
the following table.
x 0 ¼=6 ¼=3 ¼=2 2¼=3 5¼=6
f(x) 10 12 15 20 17 11
BOUNDARY VALUE PROBLEMS
PART-A
1. Classify the partial di(R)erential equation
(x + 1)zxx + p2(x + y + 1)zxy + (y + 1)zyy + yzx ¡ xzy + 2 sin x = 0:
2. Write down all possible solutions of one dimensional wave equation.
3. A taut string of length 50 cm fastened at both ends, is disturbed
from its position of equilibrium
by imparting to each of its points an initial velocity of magnitude kx
for 0 < x < 50. Formulate
the problem mathematically.
4. Write down all possible solutions of the one dimensional heat °ow equation.
5. If the temperature at one end of a bar, 50 cm long and with
insulated sides, is kept at 0±C and
that the other end is kept at 100±C until steady state conditions
prevail, ¯nd the steady state
temperature in the rod.
PART-B
6. A tightly stretched string with ¯xed end points x = 0 and x = 50 is
initially at rest in its
equilibrium position. If it is set to vibrate by giving each point a
velocity v = v0 sin ¼x
50
cos
2¼x
50
,
¯nd the displacement of the string at any subsequent time.
7. A tightly stretched string with end points x = 0 and x = L is
initially in a position given by
y(x; 0) = kx. If it is released from this position, ¯nd the
displacement y(x; t) at any point of the
string.
8. A rod 30cm. long has its ends A and B kept at 20±C and 80±C,
respectively, until steady state
conditions prevail. The temperature at each end is then suddenly
reduced to 0±C and kept so.
Find the temperature function u(x; t) taking x = 0 at A.
2
9. A rod of length l has its ends A and B kept at 0±C and 100±C
respectively until steady state
conditions prevail. If the temperature of A is suddenly raised to 50±C
and that of B to 150±C,
¯nd the temperature distribution at any point in the rod.
10. A rod of length 30cm has its ends A and B kept at 20±C and 80±C
respectively until steady state
conditions prevail. If the temperature of A is suddenly raised to 40±C
while that the other end B
is reduced to 60±C, ¯nd the temperature distribution at any point in the rod.
LAPLACE TRANSFORMS
PART-A
1. Find the Laplace transform of f(t) = te¡t.
2. State and prove the scaling property of Laplace transform.
3. Find the inverse Laplace transform of F(s) = s
(s + a)2 + b2 :
4. State the convolution theorem of Laplace transform.
5. State the initial and ¯nal value theorem of Laplace transform.
PART-B
6.(i) Find the Laplace transform of the function f(t) =
½
t; 0 < t < 3
3; t > 3 .
(ii) Find the Laplace transform of the periodic function
f(t) =
½
a; 0 · t < a
¡a; a · x · 2a
and f(t + 2a) = f(t).
7.(i) Find the inverse Laplace transform of
3s + 7
s2 ¡ 2s ¡ 3
.
(ii) Solve the di(R)erential equation y00 ¡ 2y0 ¡ 8y = 0, y(0) = 3 and
y0(0) = 6 using Laplace transform.
8.(i) Using convolution theorem, ¯nd L¡1
½
16
(s ¡ 2)(s + 4)
¾
.
(ii) Solve the initial value problem y00 ¡ 6y0 + 9y = t2e2t, y(0) = 2
and y0(0) = 6, using Laplace
transform.
9.(i) Prove that Lff00(t)g = s2F(s) ¡ sf(0) ¡ f0(0) where F(s) = Lff(t)g.
(ii) Find the inverse Laplace transform of
2
(s + 1)(s2 + 1)
.
10.(i) Find the Laplace transforms of f(t) = e4t cosh 5t.
(ii) Verify the initial value theorem for the function f(t) = 5 + 4 cos 2t.
(iii) Solve x0 = 2x ¡ 3y; y0 = y ¡ 2x, x(0) = 8, and y(0) = 3.
FOURIER TRANSFORMS
PART-A
1. State the Fourier integral theorem.
2. Find the Fourier transform of f(x), de¯ned as f(x) =
½
1; jxj < a
0; jxj > a
.
3. Find the Fourier sine transform of f(x) = e¡ax (a > 0).
3
4. If Fs(s) is the Fourier sine transform of f(x), prove that the
Fourier cosine transform of f0(t) is
FCff0(t)g = sFs(s) ¡
r
2
¼
f0(0).
5. Show that FCff(t) cos atg =
1
2
[Fc(s + a) + Fc(s ¡ a)], where Fc(s) is the Fourier cosine transform
of f(x).
PART-B
6.(i) Find the Fourier integral representation of f(x) =
½
0; x < 0
e¡x; x > 0
(ii) Find the Fourier transform of f(x) = e¡ajxj, a > 0.
7. Find the Fourier transform of f(x) =
½
1 ¡ x2; jxj < 1
0; jxj > 1 . Hence evaluate
Z
1
0
µ
x cos x ¡ sin x
x3
¶
cos x
2 dx.
8.(i) Evaluate
Z
1
0
dx
(x2 + a2)(x2 + b2)
using transform methods.
(ii) Show that (1) Fs[xf(x)] = ¡
d
ds
Fc(s) and (2) Fc[xf(x)] = d
ds
Fs(s).
9. Find the Fourier transform of f(x) =
½
a ¡ jxj; jxj < a
0; jxj > a > 0 and hence evaluate
Z
1
0
µ
sin x
x
¶2
dx.
10. Find the Fourier sine and cosine transform of xn¡1.
¡End¡
4
MA034 - RANDOM PROCESSES
QUESTION BANK
PROBABILITY AND RANDOM VARIABLES:
PART A
1. Suppose that 75% of all investors invest in traditional annuities
and 45% of them
invest in the stock market. If 85% invest in the stock market and / or
traditional
annuities, what percentage invests in both?
2. A factory produces its entire output with three machines. Machines
I, II and III produce
50%, 30% and 20% of the output, but 4%, 2% and 4% of their outputs are defective
respectively. What fraction of the total output is defective?
3. Find the moment generating function of a random variable X which is uniformly
distributed over (-2.3) and hence find its mean.
4. If A and B ate events such that ( ) 3, ( ) 1, ( )
4 4
P A∪B = P A∩B = P A 2 .
3 =Find (
)
|
P
A
B
5. Suppose that for a RV X, 1,2,3....Calculate its moment generating
function.
E⎡⎣Xn⎤⎦=2n, n=
PART B
6. (i) Let X be a continuous RV with pdf ( ) 2 ,1 2. Xf x x
x
= 2 < < Find E[logX] (4)
(ii) The average IQ score on a certain campus is 110. If the variance of these
scores is 15, what can be said about the percentage of students with an IQ above
140 ? (6)
(iii) The MGF of a RV X is what is the MGF of Y=3X+2. Also find the mean
and variance of X. (6)
( )2⋅3et +⋅7 ,
7. (i) If the continuous RV X has pdf ( ) ( ) 1 , 1 2
9
0 otherwi
X
x
f x
2
se
x ⎧ + − < < ⎪ =⎨⎪⎩
, find the pdf
of Y=X2. (4)
(ii) If the probability that an applicant for a driver's license will
pass the road test
on any given trial is 0.8, what is the probability that he will
finally pass the test
(a) on the fourth trial (b) in fewer than four trials ? (8)
(iii) The cumulative distribution function for a RV X is given by
F (x) =1−e−3x, x≥0, find Var(3X+2). (4)
1
X
8. (i) Let X be an exponential RV with parameter λ = . Use Chebyshev's
inequality,
to find P{−1≤X≤3}.Also, find the actual probability. (6)
(ii) Let X be a continuous RV with pdf ( ) 1, 2 2
X 4 f x = − <x< .Find P{X >1}
and P{2X +1>2} 6)
(iii) Let X be a RV with the pdf
(
( ) ( 2) .
1 Xf x x
π x
= 1 − ∞ < < ∞
+
Find the pdf of
Z = tan−1 X. (4)
9. (i) The time that it takes for a computer system to fail is
exponential with mean
(ii) The Pap test makes a correct diagnosis with probability 95%.
Given that the test is
(iii) Experience has shown that while walking in a certain park, the time X, in
700 hours. If a lab has 20 such computer systems what is the probability that
atleast two fail before 1700 hours of use ? (6)
positive for a lady, what is the probability that she really has the
disease? Assume that one in every 2000 women has the disease (on an
average). (5)
minutes, between seeing two people smoking has a density function of the form
( ) x, 0.
Xf x =λxe− x> Calculate the value of λ . Find the cumulative distribution
s the probability that George who has just seen a person
smoking will see another person smoking in 2 to 5 minutes? In at least
7 minutes?
(5)
function of X.What i
10. (i) Let X be a Gamma RV with parameters n and λ . Find the moment
(ii) Suppose that, on an average, a post office handles 10,000 letters
a day with a
(iii) Peter and Xavier play a series of backgammon games until one of
them wins five
(a) Find the probability that the series ends in seven games
(b) If the series ends in seven games, what is the probability that Peter wins.
generating function of X and use it to find E[X] and Var(X). (6)
variance of 2000. What can be said about the probability that this
post office will
handle between 8000 and 12000 letters tomorrow? (6)
games. Suppose that the games are independent and the probability that Peter
wins a game is 0.58.
(4)
TWO-DIMENSIONAL RANDOM VARIABLES:
PART A
1. The joint pdf of a bivariate RV (X,Y) is given by f(x,y)= kxy,
0<x<1, 0<y<1,where k
is a constant. Find the value of k.
2. The joint pdf of a RV (X,Y) is given by f (x,y)=e− y,0< ,
f(x,y) 2 x y, 0 x 1, 0 y 1.
x≤y find the
conditional cumulative distribution function of Y given that X=x.
3. Let X and Y be two independent RVs, show that Cov(X,XY)=E[Y] VarX
4. Let the jont pdf of (X,Y) be f(x,y)=2, 0<x<y<1. Find the marginal
density function of
the RV X.
5. Given that X=4Y+5 and Y=kX+4 are the lines of regression of X on Y and Y on X
respectively. Show that 0<4k<1. If k=1/16 , find the means of the two variables.
PART B
6. Two RVs X and Y have the following joint pdf = − − ≤ ≤ ≤ ≤
Find (i) Marginal pdfs of X and Y (ii) Conditional density functions
(iii) Var (X) and
Var(Y) (iv) Correlation (v) Lines of regression (16)
7.(i) What is the probability that the average of 150 random points
from the interval
(0,1) is within 0.02 of the midpoint of the interval? (8)
(ii) Let X and Y be independent (strictly positive) exponential RVs each with
parameter λ . Are the RVs X+Y and X | Y independent? (8)
f(x,y) =e− y,0< x< y< ∞
f (x)=e− x,x>0.
8. Let the joint pdf of (X,Y) be . Find r(X,Y).
9. (i) Let X and Y be independent RVs with common pdf Find the
joint pdf of U=X+Y and V=eX. (8)
X
(ii) Suppose that, whenever invited to a party, the probability that a
person attends
with his or her guest is 1/3, attends alone is 1/3, and does not
attend is 1/3. A
company invited all 300 of its employees and their guests to a Christmas party.
What is the probability that atleast 320 will attend? (8)
10. (i) The joint pdf of (X,Y) is ( , ) , 0,0 2 [ |2 | 1]
2
f x y = ye− xy x> < y< find E eX Y = (8)
(ii) If X and Y are continuous RVs with joint pdf
( , ) 2 2,0 1,0 1
2
f x y = xy+3 y <x< < y< , find the conditional pdfs X and Y. (8)
RANDOM PROCESSES:
PART A
1. Given the random process x(t) = cos at + B sin at where a is a
constant and A and
B are uncorrelated zero-mean RVs having different density functions but common
variance σ 2 . Is x(t) wide-sense stationary?
2. Show that a Binomial process is a Markov process.
3. Prove that the sum of two independent Poisson process is a Poisson process.
4. Define (i) Ergodic process (ii) Weakly stationary random process.
5. Show that the interarrival times of a Poisson process with
intensity λ obeys an
exponential probability distribution.
PART B
6.(i) Consider the random process x(t) = cos(t +φ), φ is a RV with pdf
( ) 1 ,
2 2
f x φ
π π
( ) sin cos
φ
π
= − < < check whether x(t) is first order stationary. (8)
(ii) A stochastic process is described by xt =A t+B twhere A and B are
independent RVs with zero means and equal variances. Find the variance and
covariance of the given process. (8)
7. (i) Let N(t) be a Poisson process with parameter λ . Determine the
coefficient of
correlation between N(t) and N(t +τ > 0) ; t>0 , τ > 0 .
ii) Let {x(t), t ≥ 0} be a Poisson process with parameterλ . Suppose
each arrival is
registered with probability p independent of other arrivals. Let
{y(t), t ≥ 0} be the
process of registered arrivals. Prove that Y(t) is a Poisson process with
parameter λ p . (6)
f ( ) 1;0 1 A 8.(i) Prove that the process x(t)=8+A with α = <α < is
(a) first order
stationary (b) Second-order stationary (c) strictly stationary (d) not
ergodic. (8)
(ii) x(t) is a random telegraph type process composed of pulses of
heights +1 and -1
respectively. The number of transactions of the t-axis in a time 2 is given by
( ) 4 4 .
!
e− K
x(t) cos(t )
P k transitions
K
= Classify the above process. (8)
9. Show that the random process = +φ where φ is uniformly distributed in
(0, 2π) is (a) first order stationary (b) stationary in wide sense (c)
Ergodic. (16)
10.(i) Let {x(t), t ≥ 0}
x(0) 0
be a random process with stationary independent increments,
and assume that cov( ( ), ( )) 2 min( , ) 1 = . Show that x t x s =σ t s
σ 2
( ) cos
, where
= Var(x(1)). (8) 1
(ii) Classify the random process xt =A ωt where A and ω are RVs with joint
pdf 1
A 8 f ω (α ,β )= , 0<α <2, 8<β <12. (8)
CORRELATION FUNCTION:
PART A
1. Statistically independent zero-mean random processes x(t) and y(t) have
autocorrelation functions ( ) xx R ( ) cos 2 yy R e τ = −τ and τ πτ
respectively. Find the
autocorrelation function of the sum Z(t)=x(t)+y(t).
=
2. Suppose that a random process is wide sense stationary with
autocorrelation function
( ) 2
xx R e
τ
τ − = Find the second moment of the random variable x(5) –x(3)
3. Show that (0) 2 xx R = x .
4. Show that the autocorrelation function ( ) xx R τ is maximum at τ = 0 .
5. If x(t) is a stationary random process having mean value E[x(t)]=3 and
autocorrelation function ( ) 10 xx R 3e τ = + −τ . Find the variance of x(t).
PART B
6. Given the random process ( ) ( ) n
n
x t A g t nb φ
∞
=−∞
= Σ − + where
<t<b
, ' i A s
and
⎧1 0
( )
0
g t
otherwise
=⎨⎩
φ are independent RVs with density functions ( ) 1 ( 1) 1δ(α 1)
A 2 2 f α = δ α− + + and
f ( ) 1,0 α <b. The joint probability mass function of Ai is given by
b φ α = <
i 1 A+
i A -1 +1
-1 0.2 0.3
+1 0.3
0.2
Find the autocorrelation function Rxx (τ ) in 0 <τ < b for the above process.
7. Given ( ) ( ) k
k
xn= ΣXδn−k where the joint mass function for the RVs k X is given
below:-
k 1 X +
k X 0 1
0 0.2
0.3
1 0.3
0.2
Find the autocorrelation function ( ) xx R k and covariance function for
( ) xx L k
k = 0,1, 2,3.
8. Consider the random process ( ) ( ) k
k
xn Xδn
=−∞
= Σ −k where the ' k
∞
X s are characterized
by the joint mass function
k 1 X +
k X
0 1
0 0.5
0.1
1 0.1
0.3
Find the autocorrelation function ( ) xx R k and covariance function
for k = 0,1, 2,3. for
the above process.
9. Given the random process ( ) ( ) n
n
x t A g t nb φ
∞
=−∞
= Σ − + where
1 0 t b
( )
0
g t
otherwise
< <
and
the ' n A s and
⎧
=⎨⎩
φ are independent RVs with density functions f ( ) 1, 0 b.
b φ α = <α < and
( ) 1 ( 1) 1δ(α+1) Evaluate the autocorrelation function
Ai 2 2 f α = δ α− + ( ) xx R τ in
0 <τ < b for the above process. Also the joint probability mass
function for the ' i A s is
given by
i 1 A+
i A -1 +1
-1
1
4
1
4
+1
1
4
1
4
10. Given the random process ( ) ( ) n x t A g t nb φ
∞
−∞
= Σ − + where ' n A s
and
1 0
( )
0
g t
otherwise
⎧
=⎨⎩ <t<b
φ are independent RVs with density functions ( ) 1 ( 1)+1δ(α 1)
Ai 2 2 f α = δ α− +
and ( ) 1,0 α <b. Evaluate the autocorrelation function ( ) xx f R
b
α = < φ τ in 0 <τ < b .
The joint probability mass function of i A is given by
i 1 A+
i A -1 +1
-1
1
3
1
6
+1
1
6
1
3
SPECTRAL DENSITIES:
PART A
1. State and prove any one of the properties of cross spectral density
functions.
2. The autocorrelation function of a random process x(t) is ( ) 3 2 4
2 xx R τ = + e τ
( )
− . Find the
power spectral density of x(t).
3. A widesense stationary noise process N(t) has an autocorrelation function
R Pe , τ = −ϑ τ −∞ NN <τ<∞ with P as a constant. Find its power
density spectrum.
4. The power spectral density of a stationary random process is a constant in a
symmetrical interval about zero and zero outside the interval. Compute the
autocorrelation function.
5. Which of the following functions could be a power spectral density?
(i) b+ f2
a (ii) 2
a
f − b
PART B
6. (i) Consider two independent zero-mean random processes x(t) and
y(t) with power
spectral densities and respectively. Define new random processes
z(t)=x(t)+y(t), x(t)= x(t)-y(t) and
S (jw) S (jw) xx yy
ω(t)=x(t)y(t). Find formulas for, , and
. (8)
S (jw) S (jw)
S (jw)
zz uu
ωω
(ii) Given the power spectral density
1+ω 2
ω4+4ω2+4
. Use residue theory to find the average
power in the process x(t). (8)
7.(i) Two random processes x(t) and y(t) are given by x(t)= Acos(ωt+θ )
y(t) Asin( t )
and
= ω +θ where A and ω are constants and θ is a uniform RV over (0, 2π ) .
Find the cross-spectral density functions S ( ) xy ω and ( ) yx S ω and verify
( ) xy S ω = ( ) yx S −ω .
(ii) Find the cross-correlation function corresponding to the
cross-power spectrum
( ) (9 2) (3 )2 xy S
j
ω 6
ω ω
=
+ + +
.
8.(i) The autocorrelation function of a signal is
2
e 2k2
τ
−
where k is a constant. Find the power
spectral density and average power. (8)
(ii) Show that in an input-output system the energy of a signal is
equal to the energy of its
spectrum. (8)
9.(i) Define convolution and correlation integrals for an input-output
system. State and prove
Wiener-Khinchine theorem. (8)
(ii) A random process x(t) is given by x(t) =bcos(ωt +θ ) were θ is a
RV, b and ω are
constants. Find the autocorrelation and power spectral density functions. (8)
10. (i) For a linear system with random input x(t), the impulse
response h(t) and output y(t),
obtain the cross correlation function and cross power spectral density
functions.
(8)
(ii) The power spectrum density function of a wide sense stationary
process x(t) is given
by ( )2 2
( ) 1
4 xx S ω
ω
=
+
. Find its autocorrelation and average power.
(8)
--End--
MA 035 DISCRETE MATHEMATICS
QUESTION BANK
LOGIC:
PART-A
1. What are the possible truth values for an atomic statement?
2. Symbolize the following statement with and without using the set of positive
integers as the universe of discourse. "Given any positive integer,
there is a greater
positive integers".
3. When a set of formulae is consistent and inconsistent?
4. What are free and bound variables in predicate logic?
5. Show that {↑}and {↓} are functionally complete sets.
PART-B
6. Without constructing truth table show that (¬P ∧ (¬Q ∧R)) ∨ (Q ∧R)
∨ (P ∧R) ⇔ R.
7. Without constructing truth table verify whether the formula
Q∨ (P ∧ ¬Q) ∨ (¬P ∧ ¬Q) is a contradiction or tautology.
8. Without constructing truth table obtain PCNF of (P → (Q∧R))∧ (¬P → (¬Q∧ ¬R))
and hence find its PDNF.
9. Use rule CP to show that
(∀x)(P(x) → Q(x)),(∀x)(R(x) → ¬Q(x))⇒ (∀x)(R(x) → ¬P(x)).
10. Use indirect method of proof to show that (∃z)Q(z) is not valid
conclusion from the
premises (∀x)(P(x) → Q(x)) and (∃y)Q(y).
COMBINATORICS
PART-A
1. Use mathematical induction to prove that , where n > 1.
2. State and prove Pigeon hole principle.
3. How many positive integers not exceeding 100 that is divisible by 5?
4. What is the minimum number of students required in discrete mathematics class
to be sure that at least six will receive the same grade, if there are
five possible
grades?
5. Find the recurrence relation for the sequence for
PART-B
6. Find the number of integers between 1 and 250 that are not
divisible by any of
the integers 2, 3, 5 and 7.
7. Write the recurrence relation for Fibonacci numbers and hence solve it.
8. Solve the recurrence relation with
9. Find the generating function of Fibonacci sequence F(n) = F(n-1) + F(n-2) for
with F(0) = F(1) = 1.
10. Solve, by using generating function, the recurrence relation
with
GROUPS
PART-A
1. Is the set N = { 1, 2, ………..} under the binary operation * defined
by x * y = max
{ x , y } semi group (or) monoid ? Justify your claim.
2. Show that the inverse of an identity element in a group G,∗ is itself..
3. If G,∗ is an abelian group show that for all a, b in G, (a ∗b)n = an ∗bn .
4. If every element in a group is its own inverse, verify whether G is
an abelian
group or not.
5. What is meant by ring with unity?
PART-B
6. Show that the set of all permutations of three distinct elements with right
composition of permutation is a permutation group. Is it an abelian group ?
7. Show that every finite group of order n is isomorphic to a
permutation group of
degree n.
8. Show that the order of a subgroup of a finite group G divides the
order of the
group G.
9. Let H be a nonempty subset of a group G,∗ . Show that H is a subgroup of G if
and only if a ∗b−1 ∈H for all a,b ∈H.
10. Show that the Kernel of a group homomorphism is a normal subgroup
of a group.
LATTICES
PART- A
1. Let N be the set of all natural numbers with the relation R as
follows: x R y if and
only if x divides y. Show that R is a partial order relation on N.
2. Draw the Hasse diagram of the set of all positive divisors of 45.
3. If the least element and greatest element in a poset exist, then
show that they are
unique.
4. If A = (1,2) is a subset of the set of all real numbers, find least
upper bound and
greatest lower bound of A.
5. Show that absorption laws are valid in a Boolean algebra.
PART-B
6. Show that in a complemented distributive lattice, the De Morgan's laws hold.
7. If L is a distributive lattice with 0 and 1 , show that each
element has atmost one
complement.
8. Show that every distributive lattice is modular. Is the converse
true? Justify the
claim.
9. Show that a lattice L is modular if and only if for all x,y,z∈L , =
(x z) .
x ∨(y ∧ (x ∨ z))
(x ∨ y) ∧ ∨
10. Which of the following lattices given by the Hasse diagrams are
complemented,
distributive and modular?
1
35
9
3
15
5
0
a
c
d e
1
b
30
15
3
1
2
6 10
5
d
c
b
e
a
(a) (b) (c) (d)
GRAPHS
PART-A
1. How many edges are there in a graph with 10 vertices each of degree 5?
2. If the simple graph G has n vertices and m edges, how many edges does
have?
3. Define regular graph and a complete graph.
4. What is meant by isomorphism of graphs?
5. Define Euler and Hamilton paths.
PART-B
6. If G is a simple graph with n vertices with minimum degree , show
that G is connected.
7. Show that if g is a self complementary simple graph with vertices,
then .
8. Verify the following graphs are isomorphic.
9. If G is a connected simple graph with n vertices ( and if the degree of
each vertex is at least n/2, then show that G is Hamiltonian.
10. For what value of n the following graphs are Eulerian
--End--
MA039 PROBABILITY AND STATISTICS
QUESTION BANK
PROBABILITY AND RANDOM VARIABLES
PART A
1. The probabilities of A, B and C solving a problem are 1=3; 2=7 and
3=8 respectively.
If all three try to solve the problem simultaneously, ¯nd the
probability that exactly
one of them will solve it.
2. A continuous random variable X has the following probability density function
f(x) =
8<
:
x2
3 ¡1 < x < 2
0 otherwise
Find the distribution function F(x) and use it to evaluate P(0 < X · 1).
3. If the random variable X has the moment generating function Mx(t) =
3
3 ¡ t
, obtain
the standard deviation of X.
4. For a binomial distribution with mean 6 and standard deviation p2,
¯nd the ¯rst
two terms of the distribution.
5. If X is a Poisson random variable such that P(X = 2) = 2
3P(X = 1), ¯nd P(X = 0).
PART B
6. (a) The probability function of an in¯nite distribution is given by
P(X = j) =
1
2j for j = 1; 2; ¢ ¢ ¢ ;1. Verify if it is a legitimate
probability mass function and also ¯nd P(Xis even), P(X ¸ 5) and
P(Xis divisible by 3). (8)
(b) If a random variable X has a pdf
f(x) =
8<
:
1
3
; ¡1 < x < 2
0; otherwise
¯nd the moment generating function of X. Hence ¯nd the mean and
variance of X. (8)
7. (a) Find the ¯rst four moments about the origin for a random
variable X having
the pdf f(x) =
4x(9 ¡ x2)
81
; 0 · x · 3. (8)
(b) In a bolt factory machines A;B;C manufacture respectively 25,35 and 40 per-
cent of the total. Of their output 5,4 and 2 percent are defective bolts respec-
tively. 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? (8)
1
8. (a) Find the moment generating function of a Poisson random
variable and hence
¯nd its mean and variance. (8)
(b) Suppose that a trainee soldier shoots a target in an independent fashion. If
the probability that the target is shot on any one shot is 0.7,
i. What is the probability that the target would be hit on tenth attempt?
ii. What is the probability that it takes him less than 4 shots?
iii. What is the probability that it takes him an even number of shots?
(8)
9. (a) Find the moment generating function of a Geometric random variable and
hence ¯nd its mean and variance. (8)
(b) The time required to repair a machine is exponentially distributed with
parameter 1=2
i. What is the probability that the repair time exceeds 2 hours?
ii. What is the conditional probability that a repair takes at least 10 hours
given that its duration exceeds 9 hours?
(8)
10. (a) State and prove the memoryless property of an exponential
distribution. (8)
(b) If X and Y are two independent random variables having density functions
fX(x) = 2e¡2x; x ¸ 0 and fY (y) = 3e¡3y; y ¸ 0, ¯nd the density function of
U = X + Y . (8)
TWO DIMENSIONAL RANDOM VARIABLES
PART A
11. The joint pdf of two random variables X and Y is given by
f(x; y) =
(
c(1 ¡ x)(1 ¡ y); 0 · x · 1; 0 · y · 1
0; otherwise
Find the constant c.
12. The joint pdf of (X; Y ) is given by f(x; y) = xy2 +
x2
8
; 0 · x · 2; 0 · y · 1. Find
P(X < Y ).
13. Let (X; Y ) be a two dimensional non-negative continuous random
variable having
the joint density function
f(x; y) =
(xy
36
x; y = 1; 2; 3
0 otherwise
If U = X + Y and V = X ¡ Y then obtain the joint pdf of U and V .
14. Let X and Y be any two random variables and a, b be constants. Prove that
Cov(aX; bY ) = ab Cov(X; y).
15. State Central Limit theorem.
2
PART B
16. (a) The joint probability density function of a random variable
(X; Y ) is given by
f(x; y) = kxye¡(x2+y2); x > 0; y > 0. Find the value of k and prove that X and
Y are independent. (8)
(b) If the independent random variables X and Y have the variances 36 and 16
respectively, ¯nd the correlation coe±cient, rUV where U = X + Y and V =
X ¡ Y . (8)
17. (a) The joint pdf of the random variable is given by
f(x; y) = e¡(x+y); for x ¸ 0; y ¸ 0:
Find the pdf of U =
X + Y
2
. (6)
(b) If the joint probability density function of (X; Y ) is given by
f(x; y) =
(
c(x2 + y2); 0 · x · 1; 0 · y · 1
0; otherwise
Find the conditional densities of X given by Y and Y given X. (10)
18. The joint pdf of two random variables X and Y is given by
f(x; y) =
(
k[(x + y) ¡ (x2 + y2)]; 0 < (x; y) < 1
0 otherwise
Show that X and Y are uncorrelated but not independent. (16)
19. Let X and Y be random variables having joint density function
f(x; y) =
(
x + y; 0 · x · 1; 0 · y · 1
0; otherwise
Find the correlation coe±cient rXY . (16)
20. The probability density function of two random variables X and Y is given by
f(x; y) =
8<
:
3
2
(x2 + y2); 0 · x · 1; 0 · y · 1
0; otherwise
Find the lines of regression of X on Y and Y on X. (16)
RANDOM PROCESSES
PART A
21. De¯ne strict sense and wide sense stationary process.
22. Prove that sum of two independent Poisson process is a Poisson process.
3
23. A fair dice is tossed repeatedly. If Xn denotes the maximum of the
number occurring
in the ¯rst n tosses, ¯nd the transition probability matrix P of the
Markov chain
fXng.
24. Obtain the steady state probabilities for an (M=M=1) : (N=FIFO)
queuing model.
25. State Little's formula for an (M=M=1) : (1=FIFO) queueing model.
PART B
26. (a) If fN1(t)g and fN2(t)g are two independent Poisson process
with parameter
¸1 and ¸2 respectively, show that
P(N1(t) = k=N1(t) + N2(t) = n) =
µ
n
k
¶
pkqn¡k
where p =
¸1
¸1 + ¸2
and q =
¸2
¸1 + ¸2
. (8)
(b) Let fXng be a Markov chain with state space f0; 1; 2g with initial
probability
vector p(0) = (0:7; 0:2; 0:1) and the one step transition probability matrix
P =
0
@
0:1 0:5 0:4
0:6 0:2 0:2
0:3 0:4 0:3
1
A
Compute P(X2 = 3) and P(X3 = 2;X2 = 3;X1 = 3;X0 = 2). (8)
27. (a) Show that the random process X(t) = Acos(!t+µ) is a Wide Sense
Stationary
process if A and ! are constants and µ is a uniformly distributed random
variable in (0; 2¼). (8)
(b) Consider a Markov chain with transition probability matrix
P =
0
@
0:5 0:4 0:1
0:3 0:4 0:3
0:2 0:3 0:5
1
A
Find the steady state probabilities of the system. (8)
28. (a) Assume a random process X(t) with four sample functions
x(t; s1) = cos t; x(t; s2) = ¡cos t; x(t; s3) = sin t; x(t; s4) = ¡sin t which
are equally likely. Show that is is wide-sense stationary. (10)
(b) If customers arrive at a counter in accordance with a Poisson process with a
mean rate of 2 per minute, ¯nd the probability that the interval between 2
consecutive arrivals is (i) more than1 min, (ii) between 1 min and 2 min and
(iii) 4 min or less. (6)
29. (a) The probability distribution of the process fX(t)g is given by
P(X(t) = n) =
8><
>:
(at)n¡1
(1 + at)n+1 ; n = 1; 2; 3; ¢ ¢ ¢
at
1 + at
; n = 0
Show that it is not stationary. (10)
4
(b) Arrivals at the telephone booth are considered to be Poisson with an average
time of 10 min. between one arrival and the other. The length of the phone call
is assumed to be distributed exponentially with mean 3 min.Find the average
number of persons waiting in the system. What is the probability that a person
arriving at the booth will have to wait in the queue? (6)
30. In a railway marshalling yard, goods trains arrive at a rate of 30
trains per day.
Assuming that the inter-arrival time follows an exponential distribution and the
service time is also exponential with an average of 36 minutes. Find
(a) the mean queue size
(b) the average waiting time in the system
(c) the average number of trains in the queues
(d) the average waiting time in the queue
(e) the probability that the queue size exceeds 10
(16)
RELIABILITY ENGINEERING
PART A
31. Four units are connected with reliabilities 0:97; 0:93; 0:90 and
0:95. Determine the
system reliability when they are connected (i) in series (ii) in parallel.
32. Reliability of the component is 0.4. Calculate the number of component to be
connected in parallel to get a system reliability of 0.8.
33. The reliability of a component is given by R(t) =
µ
1 ¡
t
t0
¶2
; 0 < t < t0, where t0
is the maximum life of the component. Determine the hazard rate function.
34. A fuel pump with an MTTF of 3000 hours is to operate continuously
on a 500 hour
mission. Determine the reliability.
35. A computer has a constant failure rate of 0.02 per day and a
constant repair rate of
0.1 per day. Compute the interval availability for the ¯rst 30 days
and the steady
state availability.
PART B
36. (a) The density function of the time to failure(in years) of a
component manufac-
tured by a certain company is given by f(t) =
200
(t + 10)3 ; t ¸ 0.
i. Derive the reliability function and determine the reliability for the ¯rst
year of operation.
ii. Compute the MTTF.
iii. What is the design life for the reliability of 0.95?
5
(8)
(b) The time to repair a power generator is denoted by its pdf
m(t) =
t2
333
; 1 · t · 10 hours:
i. Find the probability that the repair will be completed in 6 hours.
ii. What is the MTTR?
iii. Find the repair rate.
(8)
37. (a) Given that R(t) = e¡p0:001t; t ¸ 0,
i. Compute the reliability for a 50 hour mission.
ii. Find the hazard rate function.
iii. Given a 10 hour warranty period, compute the reliability for a 50 hour
mission.
iv. What is the average design life for a reliability of 0.95, given a 10 hour
warranty period?
(8)
(b) A critical communication relay has a constant failure rate of 0.1
per day. Once
it has failed the mean time to repair is 2.5 days. What are the point
availability
at the end of 2 days, the interval availability over a 2 day period
and the steady
state availability. (8)
38. (a) A component is found to have its life exponentially
distributed with a constant
failure rate of 0:03 £ 10¡4 failures per hour
i. What is the probability that the component will survive beyond 10,000
hours?
ii. Find the MTTF of the component.
iii. What is the reliability at the MTTF?
iv. How many hours of operation is necessary to get a design life of 0.90?
(8)
(b) Discuss the reliability of a two component redundant system with
repair using
Markov analysis. (8)
39. (a) Find the variance of the time to failure for two identical
units, each with a fail-
ure rate ¸. placed in standby parallel con¯guration. Compare the results with
the variance of the same two units placed in active parallel
con¯guration. (8)
(b) Six identical components with constant failure rates are connected
in (i) high
level redundancy with 3 components in each sub system (b) low level re-
dundancy with 2 components in each subsystem. Determine the component
MTTF in each case to provide a system reliability of 0.90 after 100 hours of
operation. (8)
6
40. (a) Find the reliability of the system diagrammed below
0.95 0.99
0.7
0.7
0.7
0.75
0.75
0.9
H
O
F
G
C
D
E
A B
(8)
(b) The density function of time to failure of an appliance is f(t) =
32
(t + 4)3 ; t > 0.
Find the reliability function R(t), the failure rate ¸(t) and the MTTF. (8)
DESIGN OF EXPERIMENTS AND QUALITY CONTROL
PART A
41. What are the basic principles of experimental design?
42. Describe Latin Square Design.
43. Depict the ANOVA table for two way classi¯cation.
44. The data given below are the number of defectives in 10 samples of
100 items each.
Sample No. 1 2 3 4 5 6 7 8 9 10
Number of defects 6 16 7 3 8 12 7 11 11 4
Construct a p-chart and comment on the nature of the process.
45. The following data gives the number of defects in 15 pieces of
cloth of equal length
when inspected in a textile mill.
Sample No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Number of defects 3 4 2 7 9 6 5 4 8 10 5 8 7 7 5
Construct a c-chart and comment on the nature of the process.
PART B
46. In order to determine whether there is signi¯cant di(R)erence in
the durability of 3
makes of computer, samples of size 5 are selected from each make and
the frequency
of repair during the ¯rst year of purchase is observed. The results
are as follows:
7
A 5 6 8 9 7
B 8 10 11 12 4
C 7 3 5 4 1
Test whether there is signi¯cant di(R)erence in the durability of the
3 makes of the
computers. (16)
47. Three machines A, B, C gave the production of pieces in four days as below:
A 17 16 14 13
B 15 12 19 18
C 20 8 11 17
Is there a signi¯cant di(R)erence between machines? (16)
48. Yields of four varieties of paddy in three blocks are given in the
following table
Farmers I II III
A 10 9 8
B 7 7 6
C 8 5 4
D 5 4 4
(a) Is the di(R)erence between varieties signi¯cant?
(b) Is the di(R)erence between blocks signi¯cant?
(16)
49. Four farmers each used four types of manures for the crop and
obtained the yield
(in quitals) as below:
Farmers 1 2 3 4
A 22 16 21 12
B 23 17 19 13
C 21 14 18 11
D 22 15 19 10
Is there any signi¯cant di(R)erence between (i) farmers (ii) manures? (16)
50. Analyze the variance in the Latin square of yields of wheat where
P,Q,R,S represent
the di(R)erent manures used.
S222 P221 R223 Q222
Q224 R223 P222 S225
P220 Q219 S220 R221
R222 S223 Q221 P222
Test whether the di(R)erent manures used have given signi¯cantly
di(R)erent yields.
(16)
8
MA040 PROBABILITY AND QUEUING THEORY
QUESTION BANK
PROBABILITY AND RANDOM VARIABLES
PART-A
1. If A and B are independent events with P(A)= 2
1 and P(B) = 3
1 , find P(A ∩ B)
and P(A ∩ B).
2. In a community, 32% of the population are male smokers; 27% are
female smokers.
What percentage of the population of this community smoke?
3. A discrete random variable has moment generating function Mx(t) = e
2(e t -1).
Find E(X) and P(X=2).
4. For exponential random variable X, prove that P(X> x+y) / X>x) = P(X>y).
5.
If a random variable X has the probability density function f(x)=
⎪⎩
⎪⎨
⎧ <
0, otherwise,
, | | 2
4
1 x
find P(X<1) and P(2X+3>5).
PART- B
6. (i) Let A and B be two independent events. It is known that P(A B)=0.64 and
P(A B)=0.16. Find P(A) and P(B).
∪
∩
(ii) A continuous random variable X has probability density function
f(x)= . Find the value of constant C. Obtain the moment
generating function of the random variable X and hence obtain its mean and
variance.
⎩ ⎨ ⎧
≤
>
0, x 0
cxe-2x , x 0
(iii) A random variables X has the probability density function
f(x) = e −2| x | , - ∝ < x ∝ . If Y = X , find the probability density
function of Y
and P (Y < 2).
2
7. (i) A bag contains 3 black and 4 while balls. 2 balls are drawn at
random one at a
time without replacement.
(a) What is the probability that a second ball drawn is white.
(b) What is the conditional probability that first ball drawn is white
if the second
ball is known to be white?
(ii) Let X be a exponential random variable with mean 1. Find the probability
density function of Y= -loge X and E(Y).
(iii) A random variable X has a mean of 4 and a variance of 2. Use the
Chebyshev's
inequality to obtain the upper bound.
8. (i) Three machines A, B and C produce identical items of their
respective output
5%, 4% and 3% of the items are faulty. On a certain day A has produced
25%, B has produced 30% and C has produced 45% of the total output. An
item selected at random is found to be faulty. What are the chance that it was
produced by C?
(ii) A test engineer discovered that the cumulative distribution function of the
lifetime of an equipment in years is given by F (x)= .
0, x 0
1 5 , 0
1
⎪⎩
⎪⎨
⎧
<
− ≥ − e x x
(a) What is the expected lifetime of the equipment?
(b) What is the variance of the lifetime of the equipment?
(c ) Find P(X>5) and P(5< X < 10).
(iii)
If a random variable X has the probability density function f(x) =
2
1 e ,
- ∝ < x ∝ , find the M.G.F of X and hence obtain its mean.
−| x |
9, (i) Suppose that, for a discrete random variable X, E(X) = 2 and
E(X(X-4)) = 5.
Find the variance and standard deviation of -4X + 12.
(ii) Let X be a geometric random variable with parameter p.
(a) Determine the moment generating function of X.
(b) Find the mean of X for p = 2/3.
(c) Find P(X>10) for p=2/3.
(iii) Message arrive at a switchboard in a Poisson manner at a average
rate of six
per hour. Find the probability for each of the following events:
(a) Exactly two messages arrive within one hour.
(b) At least three messages arrive within one hour.
10. (i) A carton 24 hand grenades contains 4 that are defective. If three hand
generates are randomly selected from this carton, what is the probability that
exactly 2 of them are defective?
(ii) Each sample of water has a 10% chance of containing a particular organic
pollutant. Assume that the samples are independent with regard to the presence
of the pollutant.
(a) Find the probability that in the next 18 samples, exactly 2
contain the pollutant.
(b) Determine the probability that at least four samples contain the pollutant.
(c) Find the expected number of pollutant.
(iii) Let X be a random variable with probability density function
f(x)=
⎪⎩ ⎪⎨⎧ ≤
≤
−
0, otherwise.
, 1 1
2
1 x If Y=Sin
2
πX
, find probability density function of Y.
TWO DIMENSIONAL RANDOM VARIABLES
PART- A
1.
If the joint probability density function of (X,Y) is f(x,y) =
⎪⎩
⎪⎨
⎧
≤ ≤
≤ ≤
0, otherwise,
0 2
0 2,
,
4
1
y
x
find P(X+Y ≤ 1).
2. The joint probability mass function of random variables X and Y is given by
P(X=x, Y=y) =
⎪⎩
⎪⎨
⎧ + = =
0, otherwise.
(2 ), 1,2, 1,2
18
1 x y x y
Find the marginal probability mass functions of X and Y.
3. Show that if X=Y, then Cov(X,Y)= Var(X) = Var(Y).
4. Prove that the correlation coefficient ρ xy takes value in the range -1 to 1.
5. State the central limit theorem for independent and identically
distributed random
variables.
PART- B
6. If the joint probability density function of X and Y is given by
g(x,y) = Find P(X>Y).
⎩ ⎨ ⎧
− + ≥ ≥
0, otherwise.
e (x y) , x 0, y 0
7. (i) Let X and Y have the joint probability mass function
Y X 0 1 2
0 0.1 0.4 0.1
1 0.2 0.2 0
(a) Find P(X+Y>1).
(b) Find the marginal probability mass function of the random variable X.
(c ) Find P(X=x / Y=0).
(d) Are X and Y independent random variables? Explain.
(ii) (X1 , X2) is a random sample from a population N(0,1). Show that the
distribution of X1
2 + X2
2 and
2
1
X
X
are independent and write down the
probability density functions.
8 (i) The joint probability density function of random variables X and
Y is given
by
f(x,y)=
⎩ ⎨ ⎧≤ ≤
≤
0, otherwise.
Cxy2 , 0 x y 1
(a) Determine the value of C.
(b) Find the marginal probability density functions of X and Y.
(c) Calculate E(X) and E(Y).
(d) Find the conditional probability density function of X given Y=y.
(ii) The joint probability mass function of a bivariate random variables (X,Y)
is given by P(X=0, Y=0)=0.45, P(X=0, Y=1)=0.05, P(X=1, Y=0) =0.1,
P( X=1, Y=1)=0.4. Find the correlation coefficient of X and Y.
9. (i) The joint probability density function of a bivariate random
variable (X, Y)
is given by f(x,y)=
⎩ ⎨ ⎧
< < < <
0, otherwise,
Kxy, 0 x 1,0 y 1
where K is constant.
(a) Find the value of K.
(b) Are X and Y independent?
(c) Find P(X+Y<1) and P(X>Y).
(ii) Let X and Y be positive independent random variable with the identically
probability density function f(x)=e-x , x>0. Find the joint probability
density function of U= X+Y and V=
Y
X . Are X and Y independent ?
10. (i) Let the conditional probability density function X given that Y=y be
f(x/y) =
⎪⎩
⎪⎨
⎧ > >
+
+ −
0, otherwise.
, 0, 0
1
x y e x y
y
y
Find
(a) P(X<1/ Y=2).
(b) E(X/Y=2).
(ii) The joint probability density function of random variables X and Y is given
as
f(x,y) =
⎩ ⎨ ⎧
≤ ≤ ≤
0, otherwise.
2, 0 y x 1
(a) Calculate the marginal probability density functions of X and Y
respectively.
(b)
Compute P(X<
2
1 ), P(X<2Y) and P(X=Y) .
(c) Are X and Y independent random variables ? Explain .
11. (i) Verify the central limit theorem for the following i.i.d
random variables:
For i =1,2,3,... Xi =
⎪⎩
⎪⎨
⎧
− .
2
1, with probability 1
2
1, with probability 1
(ii) The joint probability density function of X and Y is given by
f (x, y) =
⎩ ⎨ ⎧
> >
0, otherwise.
2e-x-2y , x 0, y 0
Compute
(a) P(X>1, Y<1).
(b) P(X<Y).
(c) P(X< ½).
(d) E(XY) .
(e) Cov (X,Y).
RANDOM PROCESSES
PART-A
1. Distinguish between wide-sense stationary and strict stationary processes.
2. Describe a Binomial process and hence obtain its mean.
3. Let X(t) be a Poisson process with rate λ . Find E(X(t) X (t+τ )).
4. Let X(t) = A cos 2π t, where A is some random variable. Is the
process first order
stationary? Explain.
5. Let {N(t);t ≥ 0 } be a renewal process with CDF F(t). Show that
P (N(t) = n) = F (t) - F (t) where F ( ) (t) is the n-fold convolution of F(t)
with itself.
(n) (n+1) n
PART- B
6.. (i) Consider a random process X(t) defined by X(t) = Y cosω t, t ≥ o where ω
is a constant and Y is a uniform random variable over (0,1).
(a) Describe X(t).
(b) Sketch a few typical sample functions of X(t).
(ii) Show that the time interval between successive events (or inter-arrival
times) in a Poisson process X(t) with rate μ are independent identically
distributed exponential random variables with parameter μ .
7. (i) Consider a random process X(t) defined by X(t) = Y cos(ω t+φ ) where Y
and φ are independence random variables and are uniformly distributed
over (-A, A) and (-π ,π ) respectively.
(a) Find E(X(t)).
(b) Find the autocorrelation function Rxx (t, t+τ ) of X(t) .
(c ) Is the process X(t) wide-sense stationary?
(ii) Express the answers to the following questions in terms of probability
functions.
(a) State the definition of a Markov process.
(b) State the definition of an independent increment random process.
(c) State the definition of the second order stationary process.
(d) State the definition of the strict-sense stationary process.
8. (i) Obtain the probability generating function of a pure birth and
death process
with λ and μ as birth and death rates, assuming the initial population size
as one.
(ii) Let X(t) be a Poisson process with rate λ .
Find E {(X (t) − X (s))2 } for t > s.
9. (i) Define renewal process and renewal density function. Establish the
integral equation for the renewal function.
(ii) Consider a random process X(t) defined by X(t) = U cost+ V sint,
where U and V are independent random variables each of
which assumes the values -2 and 1 with the probabilities 1/3 and 2/3
− ∝< t <∝,
respectively. Show that X(t) is wide-sense stationary but not strict-sense
stationary.
10. (i) Define Poisson process and obtain the probability distribution for that.
(ii) Consider a renewal process {N(t); t ≥ 0
≥ 0
} with an Erlang (2,1) inter-arrival
time distribution f(t) = t e −t , t . Find the renewal function
M(t) =E(N(t)) and obtain Lim t →∝
t
M(t) .
MARKOV CHAIN AND RELIABILITY
PART- A
1. Consider a Markov chain {X ; n 0,1,2,...} n = with state space
S={1,2} and one-step
transition probability matrix P = . Is state 1 periodic? If so, what is it
period.
⎥⎦
⎤
⎢⎣
⎡
1 0
0 1
2. Find the invariant probabilities (stationary probabilities) for the
Markov chain
{X ;n 0 with state space S={1,2} and one-step transition probability
matrix n ≥ }
P = ⎥⎦
⎤
⎢⎣
⎡
0 1
2
1
2
1
.
3. The hazard rate function Z(t) is given as
Z(t) =
⎩ ⎨ ⎧
− > > >
0, otherwise.
αβtβ 1, t 0,α 0,β 0
Find the reliability function and the failure time density function.
4. It is known that the cumulative distribution function of a certain system is
F(t) = 1 - e 3
− t - e 6
− t + e 2
− t where t is in years. Find the reliability function and
the MTTF for the system.
5. A component has MTBF = 100 hours and MTTR = 20 hours with both failure and
repair distributions exponential. Find the steady state availability
and nonavailability
of the component.
PART-B
6. (i) Let be a Markov chain with three states 0, 1, 2 and one-step
transition probability matrix
{X ;n ≥ 0 n }
P =
⎥ ⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢ ⎢
⎣
⎡
4
1
4
0 3
4
1
2
1
4
1
4 0
1
4
3
and the initial distribution P(Xo = i) = 1/3, i = 0, 1,2. Find
(a) P(X =2, X1 =1 / X 0 =2). 2
(b) P(X =2, X1=1 X =2). 2 0
(c) P(X =1, X =2 , X1=1, X =0). 3 2 0
(d) Is the chain irreducible Explain .
(ii) Discuss the preventive maintenance of the system and hence obtain MTTSF
for a system having n-identical units in series with exponential failure time
distribution.
7. (i) Consider the system, shown in the following figure, in which
four different
electronic device must work in series to produce a given response. The
reliability, R, of the various components are shown on the figure. Find the
reliability of the system.
(ii) Consider a Markov chain {Xn; n ≥ 0} with state space S = {0, 1}
and one-step
transition probability matrix P = .
2
1
2
1 ⎢⎣
⎡
0.9
0.9
0.97 0.95
0.9
0.9
0.9
1 0
⎥⎦
⎤
(a) Draw the state transition diagram.
(b) Is the chain irreducible? Explain.
(c) Show that state 0 is ergodic.
(d) Show that state 1 is transient.
8. (i) Discuss the reliability analysis for 2 - unit parallel system
with repair.
(ii) Consider a Markov chain {Xn; n 0} with state space S= {1,2} and one-step
transition probability matrix P =
≥
⎥ ⎥
⎦
⎤
⎢ ⎢
⎣
⎡
2
1
2
1
4
1
4
3
. Find the invariant (stationary)
probability distribution of the chain. Find P(X2 =1/ Xo =1) also.
9. (i) The life length of a device is exponentially distributed. It is
found that the
reliability of the device for 100 hour period of operation is 0.90. How many
hours of operation is necessary to get a reliability of 0.95?
(ii) Discuss the availability analysis for 2-unit parallel system with repair.
10. (i) A system has n components, the lifetime of each being an
exponential random
variable with parameterλ . Suppose that the life times of the components are
independent random variables and the system fails as soon as any of its
components fails. Find the probability density function of the time until the
system fails.
(ii) The following circuit operates only if there is a path of
functional devices
from left to right. The probability that each device functions is shown on the
graph. Assume that devices fail independent? What is the reliability of the
circuit.
0.9
0.9
0.95
0.95
0.9 0.99
QUEUING THEORY
PART A
1. In a given M| M| 1 FCFS queue, ρ = 0.5. What is the probability
that the queue
contains 5 or more customers. Find also the expected number of customers in the
system.
2. Define the effective arrival rate for M| M| 1/N FCFS queueing system.
3. Consider an M| M|C FCFS with unlimited capacity queueing system. Find the
probability that an arriving customer is forced to join the queue.
4. In an M| D|1 FCFS with infinite capacity queue, the arrival rate λ
= 5 and the
mean service time E(S) =
8
1 hour and Var(S)=0. Compute the mean number of
customers Lq in the queue and the mean waiting time Wq in the queue.
5. Using Little's formula, obtain the mean waiting time in the system for
M|M| 1/N FCFS queueing system.
PART B
6. (i) An average of 10 cars per hour arrive at a single-server drive-in teller.
Assume that the average service time for each customer is 4 minutes, and
both inter-arrival times and service times are exponential.
(a) What is the probability that the teller is idle?
(b) What is the average amount of time a driven-in customer spend in the bank
parking lot (including time in service)?
(c ) What is the average number of cars waiting in line for teller?
(d) On the average, how many customers per hour will be served by the teller?
(ii) Find the average number of customers in the M| M| 1/N FCFS queuing
system.
7. (i) Suppose that the car owners fill up when their tanks are
exactly half full.
At the present time, an average of 7.5 customers per hour arrive at a single
pump gas station. It takes an average of 4 minutes to service a car.
Assume that inter-arrival times and service times are both exponential.
(a) Compute the mean number of customers and mean waiting time in the
system.
(b) Suppose that a gas shortage occurs and panic buying takes place. So that
all car owners now purchase gas when their tanks are exactly three-quarters
full. Since each car owner is now putting less gas into the tank during each
visit to the station, we assume that the average service time has been
reduced to 3 3
1 minutes. How has panic buying affected the mean number
of customers in the system and the mean waiting time in the system.
(ii) For the M|M|C FCFS with unlimited capacity queuing system, derive the
steady-state system size probabilities. Also obtain the average number of
customers in the system.
8. (i) A one-man barber shop has a total of 10 seats. Inter-arrival times are
exponentially distributed, and an average of 20 prospective customers
arrive each hour at the shop. Those customers who find the shop full do
not enter. The barber takes an average of 12 minutes to cut each
customer's hair. Haircut times are exponentially distributed.
(a) On the average how many haircuts per hour will the barber complete?
(b) On the average, how much time will be spent in the shop by a customer
who enters?
(ii) Consider an M|G|1 queuing system in which an average of 10 arrivals
occur each hour. Suppose that each customer's service time follows an
Erlangian distribution with rate parameter1 customer per minute and shape
parameter 4.
(a) Find the expected number oasf customers waiting in line.
(b) Find the expected time that a customer will spend in the system.
(c ) What fraction of the time will the server will be idle?
9. (i) For an M|M|2 queueing system with a waiting room of capacity 5, find the
average number of customers in the system, assuming that arrival rate as 4
per hour and mean service time 30 minutes.
(ii) Consider a bank with two tellers. An average of 80 customers per hour
arrive at the bank and wait in a single line for an idle teller. The average
time it takes to serve a customer is 1.2 minutes. Assume that inter-arrival
times and service times are exponential. Determine
(a) The expected number of customers present in the bank
(b) The expected length of time a customer spends in the bank
(c ) The fraction of time that a particular teller is idle.
10. Discuss the M|G|1 FCFS unlimited capacity queueing model and hence
obtain P-K formula.