## Saturday, April 28, 2018

2015
(May)
MATHEMATICS
(Major)
Course: 601
(A: Metric Spaces and B: Statistics)
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions.
A: Metric Spaces
(Marks: 40)
1. (a) Define limit points in a metric space. 1

(b) A mapping defined by for all , where . Show that is a metric on 3
(c) Let be a metric space and . Show that (interior of ) is an open set. 2
2. (a) Define a closed set. Show that in a metric space every closed sphere in a closed set. 1+3=4
(b) Prove that in a metric space any finite intersection of open sets is open. 4
Or
Let be a metric space and . Then show that is closed if and only if contains all its limit points.
(c) Define a product metric space. 1
3. (a) Define a convergent sequence in a metric space. 1
(b) Prove that in a metric space every convergent sequence has a unique limit. 3
(c) Show that the Euclidean space is a complete metric space with the usual metric defined by Where 4
Or
Let be a Cauchy sequence in a metric space . Prove that is convergent if and only if it has a convergent subsequence.
(d) Let be a subspace of a metric space . Show that if is complete, and then is closed. 4
Or
Let be a metric space and . Then show that is dense in if intersects every non-empty open set.
4. (a) Define a continuous function in a metric space. 1
(b) Let and be metric spaces. Then prove that a function is continuous if and only if is closed in , whenever is closed in . 4
(c) Let and be metric spaces, be a continuous function and . Then show that the restriction is continuous on . 3
Or
Let and be two metric spaces and be a homeomorphism. Show that a subset is open if and only if its image is open.
5. (a) Define a compact metric space. 2
(b) Show that a closed subset of a compact metric space is compact. 3
Or
Show that a compact metric space has Bolzano-Weierstrass property.

B: Statistics
(Marks: 40)

6. (a) What is empirical probability? Mention its relation with classical probability. 2
(b) Explain two merits of axiomatic probability. Define probability measure in terms of probability function. 2
(c) Two dice are rolled. Find the probability of getting ‘an even number on the first die or a total of 8’. 3
(d) The probability of A, B and C becoming officers in a firm were respectively. The probabilities that the bonus scheme will be introduced if A, B and C becomes officers are respectively. What is the probability that A will be officer if bonus scheme has been introduced? 4
7. (a) What do you mean by dispersion? 1
(b) Find the mean and standard deviation for 542 numbers having class and frequency distribution as given below:
 Class 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90 Frequency 3 61 132 153 140 51 2

8. (a) Prove that correlation coefficient is independent of change of origin and scale. 3
(b) Calculating correlation coefficient between two variables and from, 25 pairs of observations obtained the following results:  , , , , However, later on detected that actual pairs of values    8 12
6 8
Were considered as  . Find the correct value of correlation coefficient. 4  6 14
9 6
9. (a) Explain the nature of binomial distribution.
(b) Find for a binomial variate , if and . 2
(c) Mention the importance of Poisson distribution. Find the first two central moments of Poisson distribution.      4
(d) Find the expression for probability density function of the normal distribution with mean and unit variance.    5
Or
Discuss about the chief characteristics of normal distribution and normal probability curve.
10. Discuss briefly the components of time series. Explain one of the methods used for measuring trend in time series. 6
Or
Find two monthly trend values of a factory for the month of November 2000 and September 2001 from the data given below by using principle of least squares:

 Year 1996 1997 1998 1999 2000 2001 2002 2003 2004 Profit (in crores) 12.6 14.6 18.6 14.8 16.6 21.2 18 17.4 15.8