Mathematical and Statistical Techniques

24 Pages·2015·1.18 MB·English

Mathematical and
Statistical Techniques
(As pertheRevised Syllabus of F.Y. B.Com., 2012-13, Semester I
University of Mumbai)
Dr. (Mrs.) Abhilasha S. Magar
M.Sc.(MATHS),B.Ed.,Dip.C.P.,Ph.D.
DepartmentofMathematics,
A.VartakCollege,
Vasai Road(W),Palghar –401202.
Manohar B. Bhagirath
M.Sc.(MATHS),M.Phil.,SET,PGDIT
DepartmentofMathematics,
A.VartakCollege,
Vasai Road(W),Palghar –401202.
MUMBAINEWDELHINAGPUR BENGALURUHYDERABADCHENNAIPUNELUCKNOW
AHMEDABAD ERNAKULAMBHUBANESWARINDOREKOLKATAGUWAHATI
© Authors
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PREFACE
Weare very happy to present “Mathematical andStatistical Techniques” to the students
of F.Y. B.Com. (Semester I). The book is based on the new revised syllabus. The text closely
follows thechapters, topic,patternsprescribedinthesyllabus byUniversityofMumbai.
We have tried to cover the topics in sufficient depth and in simple manner with easy to
understand examples. We have also given tutorials in the end of the unit which is convenient for
teachers aswellasstudents.
WewouldliketothankourprincipalDr.Keshav N.Ghorude,Dr.RajendraP.Deore(Professor,
Department of Mathematics, University of Mumbai) and Dr. Vinod M. Gaikwad for providing an
environmentwhichstimulatesnewthinkingandinnovation.
We would like to thank our seniormost colleague and Head, Dr. Santosh A. Shende and
also my loving friend Mrs. Arundhati Bane for the encouragement and continued support for
completingthisbook.
Special thankstoourfamilyforsupportthiswork.
Nothing is perfect in this world and there are always chances to improve. I shall always
remainobligedforanyimprovementsuggestions.
A big thanks to Mr. Srivastav and all the staff of Himalaya Publishing House Pvt. Ltd. for
bringing outthisbookintime.
Dr.(Mrs.)Abhilasha S.Magar
([email protected])
Prof.ManoharB.Bhagirath
([email protected])
SYLLABUS
Workload: Mathematics :2lecturesperweek
Statistics :3lecturesperweek
Tutorial :1perweek(perbatch)
Tutorialbatchsize :25students
SEMESTER I
Course: UBCOMFSI.6
MathematicalandStatistical Techniques-I
[A]Mathematics: (30marks)
UNITI:SHARESANDMUTUALFUNDS
(a) Shares: Concept of share, face value, market value, dividend, equity shares, preferential shares, bonus
shares.Simpleexamples.
(b) MutualFunds:Simple problems oncalculation ofnetincome afterconsidering entryload,dividend,change in
NetAssetValue(NAV)andexitload.Averagingofprice,undertheSystematicInvestmentPlan(SIP)
UNITII: PERMUTATION, COMBINATIONANDLINEARPROGRAMMINGPROBLEMS
(a) Permutation and Combination: Factorial Notation, Fundamental principle of counting, Permutation as
arrangement, Simple examples, combination as selection, Simple examples, Relation between nCr and nPr,
Examplesoncommercialapplicationofpermutationandcombination.
(b) Linear Programming Problem: Sketching of graphs of (i) linear equation Ax + By + C = 0 and (ii) linear
inequalities. Mathematical Formulation of Linear Programming Problems upto 3 variables. Solution of Linear
ProgrammingProblemsusinggraphicalmethoduptotwovariables.
[B]Statistics: (45marks)
UNITIII:SUMMARIZATIONMEASURES
(a) Measures of Central Tendencies: Definition of Average, Types of Averages: Arithmetic Mean, Median and
Mode for grouped as well as ungrouped data. Quartiles, Deciles and Percentiles. Using Ogive, locate median
andQuartiles.UsingHistogram,locatemode.CombinedandWeightedmean.
(b) Measures ofDispersions: Concept and idea ofdispersion. Various measuresRange,QuartileDeviation,Mean
Deviation,StandardDeviation,Variance,CombinedVariance.
UNITIV:ELEMENTARYPROBABILITYTHEORY
(a) Probability Theory:Concept of random experiment/trial and possible outcomes; Sample Space and Discrete
Sample Space; Events their types, Algebra of Events, Mutually Exclusive and Exhaustive Events,
Complimentaryevents.
ClassicaldefinitionofProbability,Additiontheorem(withoutproof),conditionalprobability.
(b) Random Variable:Probability distribution ofa discrete random variable; Expectation and Variance of random
variable,simpleexamplesonprobabilitydistributions.
UNITV:DECISIONTHEORY
Decision making situation, Decision maker, Courses of Action, States of Nature, Pay-off and Pay-off matrix;
Decision making under uncertainty, Maximin, Maximax, Minimax regret and Laplace criteria; simple examples to find
optimum decision. Formulation of Payoff Matrix. Decision making under Risk, Expected Monetary Value (EMV); Decision Tree;
SimpleExamplesbasedonEMV.ExpectedOpportunityLoss(EOL),simpleexamplesbasedonEOL.
PAPER PATTERN
Semester IEndExam(75marks)
Instructions(i) AllquestionsareCompulsory.
(ii) AttemptanyTHREEsub-questionsfromeachquestion.
(iii) Eachquestionisof15marks.
(iv) Simplenon-programmablecalculatorisallowed.
Q.1. (a) 5Marks
(b) 5Marks
(c) 5Marks
(d) 5Marks
Q.2. (a) 5Marks
(b) 5Marks
(c) 5Marks
(d) 5Marks
Q.3. (a) 5Marks
(b) 5Marks
(c) 5Marks
(d) 5Marks
Q.4. (a) 5Marks
(b) 5Marks
(c) 5Marks
(d) 5Marks
Q.5. (a) 5Marks
(b) 5Marks
(c) 5Marks
(d) 5Marks
CONTENTS
Chapter Title PageNo.
No.
1 SharesandMutual Funds 1–15
2 PermutationsandCombinations 16–28
3 LinearProgramming Problem (LPP)–I 29–48
[FormulationandGraphicalMethod]
4 MeasuresofCentral Tendency 49–117
5 MeasuresofDispersion 118–153
6 Probability 154–177
7 RandomVariable 178–184
8 DecisionTheory 185–229
Tutorial 230–238
UNIT I
CHAPTER
1
SHARES AND MUTUALS FUNDS
Introduction
A unit of ownership, that represents an equal proportion of a company’s capital is called as
shares.
According to financial term, the share is regarded as unit of account that can represent
several monetary instrument such as stocks, mutual funds, REIT’s etc. A stock market or share
market istheaggregation ofbuyers and sellersofstock orshares.A stock exchange isaplaceto
trade stocks. In India (i) Bombay stock Exchange (BSE) and (ii) National stock Exchange (NSE)
these arethetwostockExchange.
Shareholder: Any Person who owns at least one share of a company’s stock is called
share-holder orstockholder.
Types of shares
There aretwomajortypesofsharesare
(i) Ordinary shares: Ordinary shares are also called as equity shares or common stock,
which entitle the shareholder to share in the earnings of the company as and when they occure
andtocastavote atthecompany’sannual general metting’sandotherofficialmettings.
Equity shares are those shares which are ordinary in the course of company’s business.
Equity share capital is owned capital and every share has fixed value. Equity share are easily
transferableandalwaysirredeemable.
(ii) Preference shares: Preference shares are also called as preferred stock, which entitle
the shareholder to a fixed periodic income or interest but generally do not give him or her voting
rights.
Preferencesharesarecummulative,non-cummulative,redeemable,convertibleetc.various types
of preference shares are seen in the market based on their different structure, maturity term,
dividend rateetc.
Face value: Also called nominal value or dollar value. It is the origanal cost of the stock
shown on certificate. Thefacevalueofequityshare isnormally`10 and forPreferenceshare itis
`100.
Dividend: A share is a Part of the company. When the company makes profit, you often
receive a part of it.This isthe primarysource of income. Dividend depends on the profitmade by
a company. Dividend is declared by the company on the face value of a share. Dividends are
declared quarterly,half-yearlyorannuallydepends company’scontribution.
Bonus shares: Bonus shares are additional shares given to the shareholder without any
additional cost; based on the number of shares that a shareholder own. Normally, bonus shares
aredeclared intheratio.
Brokerage: Brokerage isthe commission charged by broker for buying and selling of shares
from buyer or seller. If you are buying shares, brokerage is added to the market price and if you
sellingshares,brokerage issubtractedfromthemarketvalue.
Some useful formulae
Totalamountinvested
1. Numberofsharespurchased=
Marketpriceof ashare
Rate of dividend  Face value  No. of shares
2. Totaldividend=
100
Netgain
3. Percentagegain= 100
AmountInvested
4. Netgain=Totalsale+Totaldividend –Totalinvestment
OR
Netgain=Totalsaleof–TotalPurchase.
Dividendreceivedper share
5. RateofReturn= 100
Investment inoneshare
Total amountinvested
6. Numberofsharessold=
Sellingpriceof ashare
Examples on shares
Example 1:Findhowmanyshares ofmarketvalue of `50withfacevalue of `10 canbe
purchased for`10,000
Solution: Given:marketvalue of1 share =`50
Face value =`10.Totalinvested amount=`10,000
Total Investment
No.ofsharesPurchased=
Market valueof1share
10,000
= =200shares
50
Example 2:Howmanyshares ofmarketpriceof `130 each, can bepurchased for` 50150
withbrokeragebeing 0.5%?
Solution: Given;marketpriceof 1share =` 130
Total Investmentamount=`50,150
Brokerage=0.5
Brokeragepershare=0.5%ofmarketprice
0.5
= 1300.65
100
Purchased priceof1share=M.V.+Brokerage
=130+0.65=130.65.
TotalInvestment
No.ofsharespurchased=
Purchasepriceof1share
50150
= =383.84 shares
130.65
Example 3:Mr.Shamowns500sharesofABCcompany.Thecompanydeclaresadividendof
10%thefacevalueofeachshareis`10.FindtotaldividendwhichMr.Shamwillget.
Solution: Given:No. ofshares =500
Rateofdividend =10%
F.V.=`10
Total dividend =Rateofdividend×F.V.×No.ofshares
10
= 10500=`500
100
Example 4:FindFacevalue of a15% share,if `12,500 wereinvested topurchase shares at
a marketvalueof `150 and total dividend of ` 125 wasreceived.
Solution: Given:Investment (Total)=` 12,500
Rateofdivident=15%
Marketvalueofeachshare=` 150
F.V.=?
TotalInvestment
No.ofshares=
M.V.of1share
12500
= =83.33shares
150
Total dividend=Rateofdividend ×F.V.×No.ofshares.
15
125= ×F.V.×83.33
100