This document contains 52 math and statistics problems assigned to students over three assignments. The problems cover topics like numerical methods, probability, interpolation, and integration. Students are asked to define terms, perform iterative calculations, derive formulas, prove statements, and evaluate integrals using techniques like the trapezoidal rule, Simpson's rule, and Newton's divided difference interpolation.

1. National Institute of Technology Rourkela
Department of Mathematics
Third Semester(Autumn), Academic Year 2015-16
Course: (MA 201) August 17, 2015
Assignment-I
1. Define the following terms. (i) rounding of a number (ii) chopping of a number
(iii) floating point form (iv)significant digits (v) errors in approximation (vi) loss of
significant digits. Explain them with suitable examples.
2. Find the number of iterations needed in bisection method to obtain an approximation
with accuracy 10−5 to the root of x3 + 4x2 − 10 = 0 lying [1,2].
3. Perform 20 iterations of bisection method to obtain a root of cos x − xex = 0.
4. Derive the Secant and Regula-Falsi methods. What is the difference between them.
Further, use these methods to determine the root of cos x − xex = 0 which lies in
(0, 1).
5. Derive the Newton-Raphson method (N-R method). Perform ten iterations to ap-
proximate (17)1/3 with the initial guess x0 = 2.
6. Find out the rate of convergence of Secant method and N-R method.
7. Apply N-R method to determine a root of cos x − xex = 0, with x0 = 1. Compare
your result with the first 20 iterations obtained by different methods.
8. Solve sin x = 10(x − 1) numerically by fixed point iteration method.
9. Solve numerically x cosh x = 1.
10. Find the smallest positive root of x3 − 5x + 3 = 0 by N-R method.
11. Find the root of sin2
x = x2 − 1 by iteration method.
12. Find the positive root of the equation
ex
= 1 + x +
x2
2
+
x2
6
e0.3x
correct to six decimal places.
13. From the equation x2 = a, deduce the Newtonian iterative procedure
xn+1 =
1
2
[
xn +
a
xn
]
for evaluation of
√
a. Use the above method to find
√
5 correct to 6-decimal places.
14. Apply N-R method to determine a root of x2 − 4x + 5 = 0. What can you conclude
about the convergence of the sequence (xn) obtained by this method.
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2. Assignment-II
15. A total of 28 percent of American male smoke cigarettes, 7 percent smoke cigars
and 5 percent smoke both cigars and cigarettes. What percentage of males smoke
neither cigars nor cigarettes?
16. two balls are randomly drawn from a bowl containing 6 white and 5 black balls.
What is the probability that one of the drawn balls is white and the drawn balls is
white and the other black?
17. A class in probability theory consists of 6 men and 4 women. An exam is given and
students are ranked according to their performance. If all rankings are considered
equally likely. What is the probability that women receive the top 4 scores?
18. If n people are present in a room, what is the probability that no two of them
celebrate their birthday on the same day of the year? How large need n be so that
this probability is less that 1
2 ?
19. An insurance company believes that people can be divided into two classes those
who are accident prone and those that are not. Their statistic show that an accident
prone person will have an accident at some time within a fixed 1-year period with
probability 0.4, whereas this probability decreases to 0.2 for a non-accident-prone
person. If we assume that 30 percent of the population is accident prone, what
is the probability that a new policy holder will have an accident within a year of
purchasing a policy? Suppose that a new policy holder has an accident within a year
of purchasing his policy. What is the probability that he is accident prone?
20. A system composed of n separate components is said to be parallel system if it
functions when at least one of the components functions. For such a system, if
component i, independent of other components, functions with probability pi, i =
1, ..., n, what is the probability that the system functions?
21. If two events A and B are independent, prove that Ac and Bc are also independent.
22. If three events A, B and C are independent then show that A is independent of
B ∪ C.
23. Using axiomatic definition of probability prove that 0 ≤ P(A) ≤ 1, for any event A.
24. Discuss demerits of the classical and frequency definitions of probability.
25. Give an example which sows that pairwise independence does not imply mutual
independent.
26. An interval of length 1, say (0,1) is divided into three intervals by choosing two points
at random. What is the probability that the three line segments form a triangle?
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3. Assignment-III
27. Let X be the number of tosses of a fair coin up to and including the first toss showing
heads. Find PX(2N), where 2N = {2n : n = 1, 2, . . . } is the set of even integers.
Also compute and sketch the distribution function FX.
28. A coin is tossed. If it shows heads, you pay 2 euros. If it shows tails, you spin a
wheel which gives the amount you win distribution with uniform probability between
0 and 10 euros. Your gain (or loss) is a random variable X. Find the distribution
function and use it to compute the probability that you will not win at least 5 euros.
29. Your car has broken down at noon. You have called a repair team, which left at
12.00 from a place 100 miles from where you are. they travel at 100 miles per hour.
In the mean time, you try to fix the car yourself. The chances of success are uniform
in the time interval between 12 and 1 O’clock with total probability 1/2. If you
succeed, you will drive to meet the repair team at 100 miles per hour. Let X be the
distance traveled by the team before they meet you. Find FX.
30. Let X have the binomial distribution B(n, p) and let Y have the binomial distribution
B(n, 1 − p). Prove that for any k = 0, 1, 2, . . . , n
PX({k}) = PY ({n − k}).
31. A safety device in a laboratory is set to activate an alarm if it register five or more
radioactive particles within one second. If the background radiation is such that the
number of particle reaching the device has the Poission distribution with parameter
λ = 0.5, how likely is it that the alarm will be activated within a given 1-second
period?
32. Let X be the random variables with normal distribution N(0, 1). Show that a+bX,
where a, b ∈ R, has the normal distribution N(a, b2).
33. Let X be a random variable with normal distribution N(0, 1). Find the density of
Y = exp(X).
34. A pair of dice are rolled. Let X be the greater and Y the smaller of the numbers
shown. Find the joint distribution and the marginal distributions of X and Y .
35. Let X and Y have the joint distribution
PX,Y ({m, n}) = 1/2(m+1)
, ifm ≥ n and 0 if m < n.
for m, n = 1, 2, . . . . Compute the marginal distribution PX and PY .
36. A friend of yours forgot the 8-digit password necessary to log into his computer.
If he tries all possible password completely at random, discarding the unsuccessful
ones, what is the expected numbers of attempt needed to find the correct password?
37. A coin is tossed repeatedly until a tail is obtained and you win ak euros, where k is
the number of heads before the first tail and a > 0. How much would you consider
a fair amount to pay to play this game?
38. On your way to work you have to drive through a busy junction, where you may be
stopped at a traffic lights. The cycle of the traffic lights is 2 minutes of green followed
by 3 minutes of red. What is the expected delay in the journey if you arrive at the
junction at a random time uniformly distributed over the whole 5-minute cycle?
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4. Assignment-IV
39. Compute f(9.2) by using Newton’s divided difference interpolation formula, from the
given data f(8.0) = 2.079442, f(9.0) = 2.197225, f(9.5) = 2.251292, f(11.0) =
2.397895.
40. Determine the formula for Trapezoidal rule and use it to estimate ln5 from
∫ 1
0
x
4 + x2
dx
with step size h = 0.1.
41. (i) Evaluate △2
[
4x2 − 25x + 31
(x − 1)(x − 2)(x − 3)
]
, with h = 1.
(ii) Establish the equality, ∇ + △ = △
∇ − ∇
△.
42. Prove the equalities:(i)△n(eax) = (eah − 1)neah, (ii)δ = ∇(1 − ∇)
−
1
2 . Here
△, ∇, and δ are the forward, backward and central difference respectively.
43. Compute f(9.2) by using Newton’s divided difference interpolation formula, from the
given data f(8.0) = 2.079442, f(9.0) = 2.197225, f(9.5) = 2.251292, f(11.0) =
2.397895.
44. Using Newton’s forward difference interpolating polynomial, obtain polynomials
P1(x), P2(x), P3(x), P4(x) and the values at x = 0.5 from the following data:
x 0 1 2 3 4
f(x) 1 5 25 125 625
. Write your conclusion. (Hints: given f(x) = 5x)
45. Using Newton’s backward difference interpolating polynomial, obtain the value of
f(0.25) from the following data:
x 0.1 0.2 0.3 0.4 0.5
f(x) 1.40 1.56 1.76 2.00 2.28
46. Estimate f(0) given that f(−2) = −29, f(−1) = 1, f(2) = 31, f(3) = 241, f(4) =
1021.
47. Using mathematical induction prove that △nxn = n!hn.
48. For the function f(x) = 3x4 − x3 + x2 + 5, find f[1, 2, 3, 7, 8]. Further show that
f[x1, x2, x3] = x1 + x2 + x3 if f(x) = x3.
49. Evaluate I =
∫ 1
0
sin(
1
1 + x2
)dx, using Simpson’s 1
3 rule using 20 sub intervals.
50. Find a polynomial of degree 3 or less which agrees with the function f(x) = x2 at
x0 = 1, x1 = 3, x2 = 6, x3 = 7. Write your conclusion.
51. Determine the formula for Trapezoidal rule and use it to estimate ln5 from
∫ 1
0
x
4 + x2
dx
with step size h = 0.1.
52. Show the trapezoidal rule is exact for polynomials of degree ≤ 1. Evaluate the
integral
∫ 1/2
0
xex2
dx with step size 0.1 by trapezoidal method.
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