1
Seat No.: Enrolment
GUJARAT TECHNOLOGICAL UNIVERSITY
PD -SEMESTER 1 (NEW SYLLABUS) EXAMINATION- WINTER 2018
Subject Code: 2910003 Date: 03-01-2019
Subject Name: Probability, Statistics and Numerical Methods
Time: 10:30 am to 01:30 pm Total Marks: 70
Instructions:
1. Attempt any five questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
Q.1 A random variable X has the density function x for x 0 . Show that
Tchebyshev's inequality gives
1
12
4
and show that the actual
probability is 3 e .
04
Let X be a random variable such that
X X X and
X X 0Determine the probability mass function of
X and the distribution of X.
03
Show that the factorial moment generating function of the Binomial
Distribution is pt . Hence or otherwise, show that
r
r n p .
04
Find the probability that at most 5 defective fuses will be found in a box of
200 fuses if experience shows that 2 per cent of such fuses are defective.
03
Q.2 The rankings of ten students in two subjects A and B are as follows:
3 5 8 4 7 10 2 1 6 9
6 4 9 8 1 2 3 10 5 7
What is the coefficient of rank correlation?
03
A factory has 3 machines B and producing 1000, 2000 and 3000 bolts
per day respectively. A produces defective, B 1.5% and C defective. A
bolt is checked at random at the end of a day and is found to be defective. What
is the probability that it came from machine
04
Fit a second degree parabola to the following data, taking x as the
independent variable
1 2 3 4 5 6 7 8 9
2 6 7 8 10 11 11 10 0
04
A coin is tossed 400 times and it turns up head 216 times. Discuss whether
the coin may be unbiased one.
03
Q.3 Random sample drawn from two countries gave the following data relating
to the heights of adult males.
Country A Country B
Mean height in inches 67.42 67.25
Standard deviation 2.58 2.50
Number in samples 1000 1200
Is the difference between the standard deviations significant?
04
2
Find a straight line for the data
50 70 100 120
12 15 21 25
03
Perform five iterations of the bisection method to obtain the root of the of the
equation 3 f x x 0
04
Find the real root of the equation 2 0 x f x xe which lies between 0.8
and 0.9 correct to three decimal places.
03
Q.4 Find the positive root of correct up to 3 decimal places using
Newton Raphson method.
04
Show that 1 1 E 03
The table gives the distance in nautical miles of the visible horizon for the given
heights in feet above the earth`s surface. Find the value of y when x 390 ft.
Height(x) 100 150 200 250 300 350 400
Distance 10.63 13.03 15.04 16.81 18.42 19.90 21.47
07
Q.5 Use Lagrange`s interpolation formula to find the value of y when x 10, if
the values of x and y are given below:
5 6 9 11
12 13 14 16
04
Evaluate
2
2
0
x e dx by Simpson's 1/3 rule with h 0.5.
03

Evaluate
5
10
1
log, x dx taking 8 subintervals, correct to four decimal places by
Trapezoidal rule.
04
Evaluate
1
2
0

1
dx
x with
1
6
h by Simpson's 3/8 rule.
03
Q.6

Solve
dy
x y
dx
with, 1 y by Euler's modified method for 0.1 x correct
to four decimal places by taking h 0.05.
07
Using Taylor series method, find y correct to four decimal places, given
that
1
3 1
dy
xy y
dx
.
04
Show that
1 1
2 2 1
2
E E



03
Q.7
Given that 2 1
dy
x y y
dx
using Rung- Kutta method find approximate
value of y(0.2) take h 0.1.
07

Solve the equation
2
2
u u
t x



subject to the condition
x,0 x 0.Carry out computations for two
levels, taking
1 1

3 36
h k .
07