Mathematical Economics PDF

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~IMl~,\ Nll r K,\l lH NTA l /GC NET Jl ITTA UGC ~IET JRF SITEcmmu.1cs )\.\. 11 5 '1easure of K ~ ~ == 3 Mesa kurtic Distribution 2 ~:= u/u/ ~ > 3 Lepto kurtic AM or Arithmetic Mean is the mean or average of the set of numbers which 2 P. - 3 is computed by adding all the terms in the set of numbers and dividing the Sllll1 Y,= r, ~ < 3 Platy kurtic by a total number of terms. 2 GM or Geometric Mean is the mean value or the central term in the set of numbers in geometric progression. The geometric mean of a geometric se- ,1easare of Skewness quence with 'n' terms is computed as the nth root of the product of all the t.erms (B is always positive) in sequence taken together. B = u 2/u ; 1 I 3 2 HM or Harmonic mean is one of the types of determining the average. Tue harmonic mean is computed by dividing the number of values in the sequence y1 ✓Bl by the sum of reciprocals of the terms in the sequence. y1 = 0 symmetric Distribution GM=../AMxHM Mode: 3M-2X > Positivity Skewed Distribution Y1 0 Yi< oNegatively Skewed Distribution If and b are postive numbers, than a+~ Arithmetic Mean (AM)= - - 2 Geometric Mean (GM)=./eh » (GM)' Harmonic Mean (HM)= a"+i = -.U.-  C\ N I A (J(; C: N l·:·1 / Hf· \ f·.T l·.CON UMI 11 7 116 ures skew ness: Measure the direction in which the value s are dispersed. It meas IP , the series. ;kgrcc of symm etry or asymmetry of. - zt_A~ ~ Dlllrlbullan lndl:ldu,1/ f{.. (l ,).i= M = Z.iM >Z ian sum of positive deviation from Med _c_o n/l~a·u, _ jif'.--(, ;,,)· For a series to be symmetrical, the the medi an. devi ation from should be equal to the sum of negative (.i = mean of X observations) Varience =a1 Properties of Mean : ve data 1. Most com mon form of average and is used only in case of Quantitati as well as ungr oupe d data. 2. Can be calculated for both grou ped CV (%) = (Sta nda rd dev iati on) X 100 3. Sum of Deviations of all items from Mean is Zero that is Mean rcx- i) = o imum. 4. Sum of Squa red Deviations from the Mean is Min tical treat men t com bine d mea n 5. It is capable of furth er math ema Propertjt'S of median i1Nl +i2N 2... L The sum of deviations from the med.ian will. be m1m mum 1f we ignore the Nl+N 2 signs AM. inter secti on of regression lines show DX-Ml=Min. 6. It can be depi cted graphically. The 2. It can be us d 7. Based on all items of the series. 3 I e in case of qualitative data can be carri ed out. t can be graphically depicted. 8. Furt her math ematical treat men t · · not based o II. 4. It is 9. GM is less than AM. n a Hems. bers , ratios and prop ortio n. Deciles => Divid. 10. GM is mos t suitable for inde x num es senes into te n equal parts. reciprocal of all items in a series. Percentiles => Divid. 11. HM is the reciprocal of AM of es senes into h ed equal. Quarti/es =>..d 0 iv1 es series int fi Undr h. ' parts. 12. It cann ot be easily show n in a grap o our equal parts  ~lMR,\ Nll t' 1-:i\l 1H NTJ\ UGC NET )lGM> HM Measures of Skewness Karllnnon lL M,ari Mode I I p Coeff. Of stewfless " Sfa:lldrd !Hviatton. 1 Coeff. Sk ~ = 9 Perfect Neg corrclatioo (-1) Where. however, the mode is till defined, this formula will be modified as under Perfect pos. correlation (+I) C.oeff. Sir_.- l.{Ne an - MecUa11)...... S.D......... Bowley (Q3+Ql-2M)/(Q3-Ql).... Correlation Low correlation No Correlation High Correlation ► Determines the relationship between two or more variables. It de Karl Pearson Coeff. of correlation termines the degree and direction of relationship between two or more than two variables. r= rcx-x)(Y-Y5° ► Simple correlation : correlation between two variables j'f. 6 P.E 'r' is significant Regression equation Y on X : Y = a+bX ~ Stmdard Error =✓--;;- 1:Y = na+b1:X 1:XY = 11:X + b1:X2 Cpper Limit : r +PE Y on X =bys= r!l'. ox Lower limit : r -PE ~=~ Cov(xy) ""=~.ix uy,i :.. Spearman's part correlation: (1904) it is useful for finding the corrdarion in case of qualitative data. 1:xy ~ or~ XonY= bxy ~ = r !oy! bxy.1:,r or Vary i'y b., = nl:XY-1:Xl:[ nl:Y1 -(l;Y) Rc;wession: Galton 1985. It is different from correlation as value of dependent Coeff. Of Determination - r2 ,·ari.able is calculated on the basis of given values of explanatory variables. Bzplaiud Variation r2 Total Variatian Properties: Coefficient of regression is independent of change in origin but not of scale. r= Jbyx.bry Coefficient of regression bxy and byx are not symmetric. If r = ± 1 Regression lines coincide If r = 0 Regression lines are right angled  ~I \ IR,\:\ II1 Ki\L IR NTA UGC NET JRF SET ECONOM ICS 123 I' ' Clustering Stratification ciusters/ Sampling strata: A........................::.....:: Census ct'nsus ta.ke5 eJdi item of the pop~ation and is considered tobe very Juthenti..: and rdiJble but it is time consuming and costly..~ 1s..:L1Un1ion of sample out of the population. Larger the sample size, B more :mthcnti..: i.:- the sample. Sampling method : C..............................,....... Probability I Random sampling ,,,-- All units of population have the same chance of being selected in the sample. D... Smwle unrestricted method : Techniques used are lottery method,card tech- _./ nique, tippet table. Stratified sa.mplin& : Population is divided into different strata. Strata are de- E.....::.....:: fine in such a way that the population inside one strata ishomogeneous in nature and outside it is heterogeneous. F.......... ,--_.......... Smematic sampling : for eg. I choose every 7th individual to be a part of my ----./, sample. Multi ~e samplin& : The sample is divided in stages. 2. Non- Probability Sampling Ouster samplin& : Same as stratified sampling but under cluster sampling, clus- All items of population don't have same chance of being selected ters are heterogeneous inside and homogeneous outside. Judgment sampling Stratified and Cluster Sampling Quota sampling Convenience sampling Stratified Cluster Population divided into Snowball sampling Population divided into few subgroups Sampling Errors : Error is the difference between sample statistic and popula- many subgroups Homogeneity within tion parameters. Heterogeneity within subgroups subgroups Heterogeneity between subgroups Homogeneity between subgroups Choice of elements Random choice of from within each subgroup subgroups  NTA UGC NET JRr SET ECONOM ICS ~l~ \R,\ NIIT Kt\ll R 124 125 I Dist.=.Given by AD Moivre, Karl gauss in 1733. Norrna..... Distribution 1s Lurntmg case of Normal Distrib ur10n. z. ·ssin Probability Distribution h poi t·es· 'The distribution is symmetrical., hence it resembles a be11- s aped 1 Binomial distribution : it is given by James Bernoulli. It is probability distribution where we have only two cases. Either a succt'SS or a failure. a kind of discrete proper urve. 1 · hh. c The two tails do not touc t e axis. ,.. Total area under normal _&__ ,.. curve is 1. Belong to mesokurtic distribution n = no. of trials ,.. MEAN= MEDIAN= MODE p = probability of success q = probability of failure r = no. of success Mean=O Measure of Kurtosis = Three SD=l BD is also called a limiting case of normal distribution Variance= 1 Mean= n·p Skewness measure = 0 SD= ✓ npq Variance = npq Poisson Distributions: it was given by Denies Poisson in 1937. It is used when It is used in the no. of trails is infinite but probability of success is very small. prob. Distribu- case of finding no. of defects, no. of accidents etc. It is discrete tion. m= mean of prob. Distribution. M (mean)= np SD= ✓np Variance == np Mean = Variance  NJ/-\ Uul.. NET JRF SET E CONOMiCS SIMRANJIT KAUR 126 'fiJ11e reversal P01 x P10 = 1 127 factor reversal test P01 x Q01 = 1¥iif¥1 tPiq -.!.. t PoIOO Critical value =l.645 II. Exactly one defective 5c 1 15c 2 Z= X: I = 11~; 100 = 4.56 20c 3 J: J30 111. Atleast one defective 1-P (no defective) 4.56 > l.645, Null Hypothesis is rejected Principal claim is right In a simultaneous toss of 2 coins, what are the possible outcomes. Find the probability of '2 head: ➔ Sample space= [ HH,TT, HT,TH] 1 P(2 heads)= - 4  NTA UGC NET JRF SET E CONOM1cs ,c progression 143 r71etr GBO arz, ar 1s a geometnc. progression 3. A ar, l\'latbematical Economics ' w11ere a is the first term and r is common rat· 10, n-1 Sequence and Series an -- ar a( 1-rn), S-- 11 ~ 1-r... der form is termed as sequence. Collection ot obJects ll1 or. ric mean of two positive numbers aand bis th r.. fh beings at different tunes form asequence. Geornet enumber -v ab For e.g. Popu.laaon o uman Toe amount of money deposited in a bank over a number of years formsa se- quence. Arithmetic mean of two positive numbers aand bis A= ~.. 2 , ll owmg Sequences 10. specific patterns are called progressions. 1.e. sum to n terms of special series Arithmetic pJogression and geo~~ sions i) 1 + 2 + 3 + ------- +n ( sum of first nnatural numbers) S = n(n+l) n 2 Series ii) 12 + 22+ 32+- - - n2 ( sum of squares of the first n Let a1, a2, a3, - - - - - - CZn be agiven sequence. natural numbers) + Then the expression a1 +a2 + a3 + - - - - U-n is called the series Sn= n(n+1)(2n +l) associated with the given sequence. Series is represented as sigma 6 notation i.e. lJ=i aK iii) 13 + 23+ 33+ - - - n3 (sum of cubes of the first n Arithmetic progression natural numbers) Let as consider an A.P with first term a and common difference d, i.e. a, Sn - [n(n+l)]z a+ d, a+ 2d, ---- 4 nth Term of the A.Pis lln =a+ ( n-l) d a=first term I = last term d =common difference n= number of terms sn =sum ton terms Theory of Sets n of A.P Sn= i [2a +(n-1) d] or Sn= la+ I]i. ll t' on must be mathematical Set is a well defined collection of objects. [This co_ec If b tiful girls does not valid] i.e. A is a set of natural numbers. Collection eau ° form a set. Two ways of expressing a set  ,1 \lh, \ \' JI l 1' \l " NTA UCC NET JRr' Str EcoNOMtcs 145 IH A is a set of natural.on of two ldl: The set of all eltmentt which are crther in set A or in set B or or TJbu lar form for eg. l}J1bl oth is called union of two !