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Special Features. Comprehensive and analytical treatment is given of all the topics. Difficult mathematical deductions have been treated logically and in a very . Fundamentals of Mathematical Statistics By S.C. Gupta – PDF Free you're little serious about your studies, you should never consider eBooks/Books in PDF. S. C. Gupta and V. K. Kapoor are Indian academicians and mathematicians with . S.C. Gupta, V.K. Kapoor Fundamentals of Mathematical Statistics a Modern Approach, 10th Edition - Ebook download as PDF File .pdf), Text File .txt) or.

Contains, besides complete theory, more than fully solved examples and more than 1, thought-provoking Problems with Answers, and Objective Type Questions V. They now take great pleilsure in presenting to the readers the ninth completely revised and enlarged edition of the book. The subjectmatter in the whole book has been rewritten in the light of numerous criticisms and suggestions received from the users of the previous editions in-lndia and abroad. These will enable 'the reader to have a better and thoughtful understanding of the basic. Test p. Many new problems are included in the exercise sets.

Repetition of questions of the same type more than what is necessary has been avoided. Further in the set of exercises, the problems have been carefully arranged and properly graded. More difficult problems are put in the miscellaneous exercise at the end of each chapter. In fact with the addition of new topics, rewriting and revision of many others and restructuring of exercise sets, altogether a new book, covering the revised syllabi of almost all the Indian urilversities, is being presfJnted to the reader.

It Is earnestly hoped that, In -the new form, the book will prove of much greater utility to the students as well as teachers of the subject. The book will also be found' of use DY the students preparing for various competitive examinations.

While writing this book our goal has been to present a clear, interesting, systematic and thoroughly teachable treatment of Mathematical Stalistics and to provide a textbook which should not only serve as an introduction to the study of Mathematical StatIstics but also carry the student on to 'such a level that he can read' with profit the numercus special monographs which are available on the subject.

The book contains sixteen chapters equally divided between two volumes. Mathematical treatment has been given to. The theory of probability which has been developed by the application of the set theory has been discussed quite in detail. A ,large number of theorems have been deduced using the simple tools of set theory. The viii simple applications of probability are also given.

The chapters on mathematical expectation and theoretical distributions discrete as well as continuous have been written keeping the'latest ideas in mind. The thirteenth and fourteenth chapters deal mainly with the various sampling distributions and the various tests of significance which can be derived from them. In chapter 15, we have discussed concisely statistical inference estimation and testing of hypothesis.

Abundant material is given in the chapter on finite differences and numerical integration. Using mode formula we get: Sometimes mode is estimated from the mean and t he median. For a symmetrical distribution, mean, median and mode coincide. If th e distribution is moderately asymmetrical, the mean, median and mode obey the fo llowing empirical relationship due to Karl Pearson: If the method of grouping gives the mod al class which does not correspood to the maximum frequency, i.

In such cases, the value of mode can be obtained by th e formula: Merits and Demerits or Mode Merits. Like median, mode can be located in some cases merely by inspe ction.

It is not always possibl e to find a clearly defined mode. In SOlD" cases, we may come across distributio ns with two modes. Such distributions are called bi-modal. If a distribution has more than two modes, it is said to be multimodal. Mode is the average to be used to find the ideal size, e. Geomet ric Mean. N i-I Merits and Demerits of Geometric Mean Merits.

U It is based upon all the-observations. C-eometric mean is used To fiild 'the rate of populati6il growth a nd the rate Of interest. Show that in finding the arithmetic mean of a set of readings on the rmometer it doe..

Then their arithmetic mean C is given by: Hence in finding the arithmetic mean of a set of n readings on a thermometer, it is immaterial whether we measure tempe ratcre in Centigrade or Fahrenheit. Harmonic Mean. Merits and Demerits of Harmonic Mean Merits. Hannonic mean is rigidly defined, based upon a ll the observations and is suitable for funher mathematical treatment.

Like geom etric mean, it is not affected much by fluctuations of sampling. It gives greate r importance to small items and is useful only when small items have to be given a greater weightage. A cyclist pedals from Iris. In this case. Rather, we have the 'following general res ult: If equal distances are covered travelled per unit of time wjth speeds equ al to V.. V2, "', V. V Proof is left as an exercise to the reader. Weighted Harmonic Mean. Instead of tixed comtant distance being with varyin g speed, let us now suppose tbat different distances, say, S.

In that c ase, the average speed is given by tbe weigbted harmonic mean oftbe speeds, the weights heing tbe corresponding distances tr-welle l, i. YOII can take a trip which entails travelling km. Wllat is YOt" average speed for tire entire distance? Speee' km.

Distance inkm. Total 15 1: Selection of an Average. A jUdicious selectio n of the ave! Since arithmetic mean satisfies all tbe properties of an ideal average as laid -down by Prof.

Yule, is familiar to a layman and furtber bas wide applic;ttions in statistical theory at large, it may be regarded as tlie best of all tbe averages. Partition Values. These arc tbe values wbicb divide tbe series into a numocr of equal parts. Frequency Distributions And Measures or Central Tendency The three points which divide b'le series into four eqqal parts are called quort iles.

The farst, second and third points are known as the r: The second qua rtile, Ql, coincides with median. The me thods of computing the partition values are the same as those of locating the me dian in the case of both discrete and continuous distributions. EX8Dlple Eig ht coins were tossed together and the number f: Calc ulate median. Cumulati ve frequency cf. Hence D. Hence Graphical Location of the Partition Values.

The partition values, Z First fonn the cumulative frequency table. Take the class inte rvals or the variate values along the x-axis and plot the corresponding cumula tive freqIJencies aJong the y-axis against the upper limit of the class inrerval or against the variare value in the case of discrete frequency distribution.

The curve obtained on joining. The graphicallucation of partition values from. Draw the cumulative fr equency curve for the follOWing distribution showing the number of marks of59 st udents in Statistics. Marks-group No. Marks-group ' No. The smooth curve obtained on joining these points is called ogive or more particularly 'less than' ogive. From 'A' draw a line p,rependicular to x-axis meeting irin ' M' say.

Then abscissa of 'M' gives the value of median. The median can also be located as follows: F rom the point of intersection of 'less than' ogive and 'more than' ogive, draw p erpendicular to OX. The abscissa of the point so obt,ained gives median. Othe r partition values, viz. What are their uses?

What are the pr inciples governing the choice o: Write sho rt notes on: It satisfies 8Imost all the requirements of a good average. The median is also an average, but it does not statisfy all the requirements of a good average. However, it carries certain merits and hence is useful in particular fields. Critically examine both the averages.

Defme ; arithmetic mean, ii geometric mean and iii hannonic mean of gro uped and ungrouped data. Compare and contrast the merits and demerits of them. S how that the geometric mean is capable of funher mathematical treatment. What are the requisites of a statisf actory average? In this light compare the relative merits and demerits of three well-mown averages. Discuss their merits. Show that i Sum of deviations about arithmetic mean is zero.

The following numbers give the w eights of 55 sbJdents of a class. Prepare a suitable frequency table. Choosing a ppropriate class-intervals, fonn a frequency table for the following data: A sam.

Fmd out the following: Age in years: Each one did his own simplification. A's method: Divide the sets into sets of ea ch, calculate the average in each set and then calculate the average of these av erages.

B's method: Divide the set into 2, and 3, numbers, take average in each set and then take the average of the averages. C's method: He averaged all other nwnbers and then added one. Are these 'methods correct? Correct, not correct, not correcL b The total sale in ' rupe es of a particular item in a shop, on to consecutive days, is reported by a cle rk as, , , , , 4.

OO, , ,, , Calculate the aver Later it was found that there was a nwnber in the machine and the reports of 4th to 8th days were more than the true values and in the last 2 days he pu t a decimal in the wrong place thus for example was really Calculate the true mean value.

Madurai Univ. If several sets of observations are combined into a single set, show that the mean of the combined set is the weighted mean of several sets. Defme the weighted arithmetic l,l1ean of a set of numbers. S how that it is unaffeci. The following table shows some data collected for the regiQlls of a country: Prove the formulae you use. Draw the Ogives and hence estimate the median. Class 32 13 Frequency 8 The following data relate to the ages of a group of workers in a factory.

The mean of the marks obtained in the same examination by a nother group of students was Find the mean of the marks obtained by bot h the groups of studel ts taken together. Find the mean of the combined distribution. What is the mean of the remaining students? The mean weight of boys in the class is 70 kilogram s and that of the girls is 55 kilograms. Find the number of boys and number of g irls in the class.

From the following data, calculate the percentage of workers getting wages a m ore than Rs. Wages Rs. Assuming that frequencies are uniformly distributed over the entire interval, a Number of persons with wages more than Rs. For the two frequency disttibutions give below the mean calculated from the first was and that from the second was Find the values of x and y.

Statistics A number of particular articles has been classified according to their weigh ts. After drying for two weeks the same articles have again been weighted and sim ilarly classified. It is known that the median weight in the frrst weighing was oz. Some frequencies a and b in the first weighing and x and y in the second are missing.

Find out the values of the missing frequencies. Frequencies Class 1Sf weighing lInd weighing Class Frequencies 1Sf weighing lind weighing 52 75 22 50 30 28 a b 11 x Y 40 Hint. Using Medi an fonnula, we shall get Now the median of 2nd weighing gives: Subtracting from , we get.

Substituting in. From the following table showing the wage distribution in a certain factory, determine: Point out the method of finding out the values o median, mode, quartiles, de ciles and percentiles graphically. Also, write down the fonnula for the computat ion of each of them for any frequency distribution. Marks No.

Find the median, the third quartile and the second decile of the distribution. Check your results by the graphical method. Age of h ead offamily. The followiflg data represent travel expenses other than transportation fo r 7 trips made during November by a salesman for a small fmn: Trip Days Expense Expense per day Rs. The salesman replied that the aver age is only Rs. The auditorrejoined that the arithmet ic mean is the appropriate measure, but that the median is Rs.

You are require d to: You take a trip whi ch entails travelling miles by train at an average speed of 60 m. What is your speed for the entire distance? What is the average speed in m. A large part of the distance is uphill and he gets a mileage of only 10 per gallon of gasoline. On the return trip he makes 15 miles per gallon.

Find the harmonic mean of his mileage. The following table shows the distribution of families according to their expenditure per week. Number ot families corresponding to expenditure groups Rs.

Expenditure 2Q No. Expenditure No. If their arithmetic mean is , find the value of x. The weights for the fll"st and the second numbers are 2 and 4 respectively.

Find th e weight of the third. The geomebic mean of 10 observations on a cer tain variable was calculated as It was later discovered that one of the obse rvations was wrongly recorded as ; fn fact it was Apply appropriate correc tion and calculate the correct geometric mean.

A variate takes the 'values a, ar, a? If A, G and H are respectively the arithmeti c mean, geometric mean and harmonic mean, show that '. Show that the mean, me dian and mode of the new distribution are given in terms of those of the first [ Kanpur Univ. Use the metho d indicated above to find the mean of the following distribution: X duration of telephone conversation in seconds , ,, , , , ,, , I respective frequency 6 28 88 Tn a frequency table, the upper boundary of each class interval has a consta nt ratio to the lower boundary.

Show that the geometric. St at. Tendency Find the minimum value of: The sum of squares of deviations is minimum when t aken from arithmetic mean and the sum of absolute deviations is minimum when tak en from median. If A, G and H be the arithmetic mean, geometric mean and har monic mean respectively of two positive numbers a and b, then prove that: When does the equality sign hold?

Designation Monthly salary in Rs. Latter i s more representative. Treating the number of letters in each word in the fo llow: It is a waste of time to apply the refined theoretical meth ods of Statistics to data which are suspect from the beginning. Match the correct parts to make a valid, statement: X2 x,, lln c i: Which measure of location win be, suitable to COl lpare: IS"; ' vii marks obtained Which of the following are true for all sets of data'?

Which of the foltowing are true in respect of a ny distribution'? Find out the missing figures: Mean - Median. Fill in blanks: Then the median strength is The value of the largest item is It was later found that it is actually Therefore, mean of these observations is If wrong o bservations are replaced: For the questions given below, give Correct answers. Vii Whenxi and Yi are two variab: Fundamentals or MadJematical Statistics. In cas e. I The harmonic mean of n n umbers is the reciprocal of the Arithmetic mean of the reCiprocals of the number s.

U For the wholesale manufactUrers interested in the type which is usually i n demand, median is the most suitable average. The average rate 'per rupee is lkg. S o for working days total attendance is 3, So the avera ge speed for the whole journey is either 35 m.

Tendency 2'43 XX In calculating the mean for grouped data, the assumption is made that tbe m ean ofthe items in each class is equal to the mid-value of the class. Be brief in your answer: Hence the production has ",eclined by percent. He drove on the first day at the rate ofA5 Ion per hour, second day at 40 km.

Which average, hannonic mean or arithmetic mean or median will give us his average speed?

Hence export trad e is not vital to the economy of that country. Is the conclusion right? It further revealed that the grandfathers of these children were also highly intelligent. Do you agree? Hence it i's safe to join military serviC'e t han to live in the city of Hyderabad. In the same year and at the same examination only out of students were successful in college Y.

Hence the teaching standard in college X was hener. It was found later th? Later on, after disbursement of salary it was discovered that the salary of two emplo yees was wrongly entered as Rs.

Their correct salaries were and 'Rs. Find the correct arithmetic mean. Is the infonnation given above'cor rect? If not, why? If we know the average alone we cannot forlll a complete idea about tbe distribution as will be cear from the following example. Consider the scries i 7.

In all tbcse cases we sce that n, tbe number of obscrvations is 5 and tbe lIIean is 9. Jllust be supported and supplemented by some other measures, One such measure is Dispersion. Literal meaning of dispersion is scatteredness'. We study dispersion to bave aJl idea ahout the bQlllpgeneit ,: In the above case we say that series i is more homogeneous less d ispersed tban the series ii Qr iii or we say that series iii is 1,IIore h eterogeneous Illore s 'anert"d thall the series i or ii , ],2.

The dcsiderdta, for an ideal measur e of dispersion arc the same as those for all ideal measure of ccnlral tendcllcy, viz. Measures 'of Dispersion. Tbe range is the.

Range is tbe simples I but a cr ude lIIeasure of dispersion. Sill ,c it is based on two extreme observations whi ch themselves are subject to 'hanee Ilul'tuations, it is not at all a n: Jiablc mcasure of. Quartile el'iation. Quartile deviation or semi-inlcrqu lIrtile rangl'.

Since mean devilltion is based on all the observations. But the step of ignoring the signs of the deviations tr, - A creates a rtificiality and ,rendcrs it useless for -further mathematical treatment. The proof is given for continuous variable in Chapter 5 The step of squaring the deviations Xi - X overcomes the drawback of ignoring the signs in mean deviation. Moreover of all the measures, standard deviation is affe cted least by fluctuations of sampling. It lIlay also be pointed out that s tandard deviittion gives greater weight to extreme values and as such has not fo.

Taking into COnSideration the pros and cons and also the w ide applications of standard deviation in statistical theory, we may regard stan dard deviation as the best and the most powerful measure of dispersion!

The squa re of standard deviation is called the variance and is given by. Relation between 0 and s. X -A , being constant is taken outside. But 'J. The same result could be obtained altematively as follows: Different J'orm ulae For Calculating Variance. S out to be in fractions, the calculation of by using 35 is ve ry cumbersome and time consuming. In order to overcome this difficulty, we shall develop different fonns of the formula 3'5 which reduce the arithmetic to a g reat extent and are very useful for com putational work.

In the following sequenc e the summation is extended over '; from 1 to n. In that. Generally the poin, in the middle of the distribution is much convenient tbough tbe formu la is true in general.

Measures of Dispersion. Skewness and Kurtosis 35 Hence variance and consequently standard deviation is independent of change of o rigin. Prove that for any discrete distri bution standard deviation is not less than mean deviation from mean.

Then We have to prove that.. Hence the result. Find the mean deviation from tire mean ant! We know that the mean of a series in A. Let Xi If;, i.. A lso l: Neglecting x;lM 3 and. Mneglecting higher powers. Hence 1. Measures of Dispersio! For a group of candidates, the mean arid standard deviation of sc ores were found to be 40 and 15 ;-espectively. Find the corrected m ean and standard deviation corresponding to tlte corrected figures.

Variance of the combined series. Ie mean oJ t Ie comb' med series. Hence from Substituting from 3' lOb and 3'10c in 3' The formula 39 can be easily generalised to the case of more than two s eries.

The first of the two samples has items -ith mean 15 a nd standard deviation 3. If the whole grollp has items with mean and stan dard deviation"; , find the standard deviation of the second grollp. Co-efficient of Dispersion. Whenever we want to compare the variability of tht WO series which differ widely in their averages or which are measured in different units, we do not merely calculate the measures of dispers ion but we calculate the co-efficients of dispersion which are pure numbers inde pendent of the units of measurement.

A -'8 1. Based upon qua nile deviation: Based upon mean deviation: Average from whkh it is calc ulated Based'upon standard deviation: Co-efficient of Variation. For comparing the variability 01 two series, we cal culate the co-efficient of variatiom; for each series. The series having greater C. Measures' of Dlspersiob.

In u'IIiill Jirm. IX6 Total waces paid AH: OI O No. Sinl'e C. Iii a The average monthly wages l,ay X,. The combined variancc 0: Jiv e and absolute measures of dispersion and describe the merits and demerits of st andard deviation. Clllicut Unlv.

April b Compute quartile deviatIOn graphicall y for the following data: Clas ses Frequency Less than 20 30 20 20 - 30 30 15 46 - 50 10 5 50 - 60 c Age distribution of hundred life insurance policyholelers is as follows: Prove that the mean deviation about the mean frequency of whose ith size X j is It is given by Number 9 16 12 26 14 12 6 5 x of the variate x, the. Mean dev. What is standard deviation? Explain its superiority over other measures of di spersion.

Calculate the mean and standard deviation of the following distribu tion: Explain clearly the ideas impJied in using arbitrary working orgin, and scale for the calculation of the arithmeti c mean and standard deviation of a frequency distribution. The arithmetic mean and variance of a set of 10 figures are known to be 17 and 33 respectively. Of the 10 figures, one figure i. Baroda U. At the time of checking it was found that one item 8 was I Incorrect. Calcu late the mean and standard deviation if '.

OJ the wrong item is omitted, and i i it is replaced by This is true of all data, b ut particularly so of numerical' dala,.. The mean of 5 observations is 44 and v ariance is It three of the five observation are 1, 2 and 6; find the' other two. Golfer A: Golfer B is more. The sum an d 'sum of squares corresponding to length X in' ems. Measures of Dispersion, Skewness and Kurtosis 4 -,6 12 13 7 19 8 - 10 5 9' 4 1 10 - 12 What is the average life of each mod el of these' refrigerators?

Which model shows more uniformity? ADS; C. Z 1 i Which firm, A ot B, pays'out larger amount as' inonthIy wages? Calculate t he. Standard deviation. Rso- A 50 80 9;. A collar manufacturer is considering the production of a new style collar to att ract young men. Nagpur Univ. Obtain the mean and standa: Obtain the mean and standard deviation of the sample of si ze obtained by combining the two samples.

Combined mean. Combined S. In a certain test for which the pass marks is 30, the distribution of marks of passing candidates clas sified by sex boys and girls were as given below' Frequency Marks Boys Girls 5 15 to 20 15 30 30 20 5 5 5 7C' Total 90 Num ber Standard Deviation Mean..

The corresponding figures for girls including the 10 failed were 35 and 9. Wiothout quetioning the propriety of this argument, su ggest. Find the mean and variance of first n-natural numbers. In a frequency distribution, the n intervals are 0 to 1, 1 to 2, Find the mean deviation and variance. If i is the mean value of all the measurements, prove that the standard deviation is In a series of measuremen..

Deihl Unlv. If the deviations are small compared with the value of the mean, show that i Mean. Froln' a. AdjustlJ ent to the variance to'correct the error is: H ons. Se Stat. The rth moment of a variable about the rt1ean given by I 1 ,. N- 'i;f; x;. We know that if d; - X; - A , then cl. Relation between"moments about mean in terms of moments about any point and v ice verso.

We have. If; ;c; - xr -.! Using , we get. Thus the rth moment of the variable x about mean is h r times the rth moment of the variable II about its mean. Sheppard's Corrections for Moments. In case o f grouped frequency distribution, while. But since the assumption is not in general true, some error, called the 'grou ping error', creeps into the calculation of the moments. Sheppard proved th at if i the frequency distribution is continuous, and U the frequency tapers off to zero in both directions, the effect due to grouping at the mid-point of the intervals can be corrected by the following fonnulae, known as Sheppard's co rrections: Charlier's Checks.

The following identitfes 'ift. Karl Pearson defined the following four c oefficients, based upon the first fout moments"about mean: The practical utility of these coefficients is discussed in and Sometiines, another coefficient based on moments,. Alpha coefficients are defined as: Factorial Moments.

Absolute Moments. The 11h absolute mom ent of the variable about the 1lean -N'I. In the usual notations. Moments about mean: Moments about the point x 2.

Skewness and Kurtosis Solution. J4 Al ' ,2 - 3III ,4 '"' 11! Obtain tlte first four moments abom the orgin, i. Comment upon tlt e nature 'of distribution. Comments of. Nature of distribution: What are Sheppard's corrections to the central moments? E stablish the relationship between the moments about mean, i.

The first three m oments of a distribution about the value 2 of the variable arc I, 16 and - Find the first four moments about t lie origin. What is the effect of change of origin and s,cale o.

The first four moments o f distribution. Obtain a s far as possible, the various characteristics of the distribution on the basis of the information given. The first four moments of a distribu tion about. Wages in ' Mean approx. Class Limits Frequency - 5 15 - 20 - 35 1 - - 10 - 5 Also apply Sheppard's corrections for moments. What must be the value of the fourth moment about the mean in order that the distribution be i leptokurtic, ii mesokurtic, and' iii platykurtic? III 0 beca use distribution is symmetrical.

Hons , ] Hint. Hence 6. Gujaratl njv. Calculate the l'orrect frequency constants. Correct Mean: OiJ1ainthe cor rect va lue of the first Jour centra I moments. If It is the magnitude of the c lass interval, tben we want: Required adjustement.. BangaIore Uhiv. Skewnes s. These are the absolute measures of ,skewness. If mode is iII-defined. M 0 moderately asymmetrical distribution. In practice, these limits are ,rarely attained. Prof Bowle3'! From Moreover skewness is positive if: Limit sIor Bowley's Coefficieitt of Skewness.

We know that for two real positive numbe rs' a and b i. Thus, Bowley's coefficient of'skewness ranges from - I to I. Furtheri we note fr om that: If Q3 Md' 4. It should be clearly understood that the r values of Jhe coe fficients of skewness obtained by Bowley's formula and Pearson's formula are not comparable, although in each case, Sk 0, implies the absence of skewness, i. It may even happen that one of them gives posi. In BowleY's coefficienl of skewness the disturbing factor of "va riation is e liininated by divi iing the absolute measur'!

The co-efficient, in is tQ be regarded aS'without sign. The skewness is positive if the larger tail of the distribution lies towards 6 P, Negatively Skewed Distribution. Karl Pearson c alls as the 'Com'exit ' of cun'e' or Kurtosis. What do you understand by skewne. Dis tinguish clearly, by giving figures, between positive and negative. Explain the methods of measuring skewness and kunosiS' of a frequency distrib ution.

Show that for any frequency distribution: Why 10 we c alculate in general,. Compute the following: Annual Sqles Rs. Find the mode of the distribution. If the sum oif the upper and lower quartiles is and median is 38, find the value of-th e upper lfnd;Jower,quartiles.

Find tbe mean and mode oflbe di stribution. Examine tbe skewness of the c. The fitst three moments about the origin 51 Kg.. Class inten'al. Deihl Univ. Which value Of 'a' gives the minimum? What will be the nev mea. Fill in the blanks: The sum o f squares of deviaJions is least When measured from I If each item is increased by 2, the median will be Therefore, the coeffident of quartile deviation is For the following questions given corr ect answers: The val ue of the fourth central moment 14 , in order that the distribution be mesokur tic should be a e9ual to 3, , b greater than 1,, c equal to 1; State which ,of the following'statements' are yure and which False.

I an e'xperimept is ,repeated under essentially homo. The pheno. Jj are kno. C' c Oh ms w, VIZ. A deterministic model is defined as a model which stipulates' t hat the co. Ii If a light ture has' lasted for I ho. It may fail to. Short History.

Bul the flfSt fo. French mathematicians, B. Pascal andP. Fermat , while. The famous 'problem of points' posed by De-Mere to Pascal is: The person who ftrst gains a certain number of pOints wins the stake. They stop playing before the game is completed.

Next stalwart in this fteld was J.. Bernoulli who se 'Treatise on Probability' was publiShed posthwnously by his nephew N. Bernoul li in De"Moivre also did considerable work in this field and p ublished his famous 'Doctrine of Chances' in Bayes Inverse probability ,P.

Laplace who after extensive re search over a number of years ftnally published 'Theoric analytique des probabili ties' in In addition 10 these, other outstanding cOntributors are Levy, Mi ses and R. Russian mathematicians also have made very valuable contrib Utions to the modem theory of probability.

Chief contributors, to mention only a few of them are,: Khintchin e law of Large Numbers and A. Kolmogorov, who axibmi. In this section we wiUdefane and explai n the various tenns which are used in the'definition of probability. Trial and E vent. The experiment is known as a triafand the outcomes are know n as events or casts.

For example: Exhaustiye Even ts. Mutually exclusive events. Equ ally likely events. Independent events. Several ev. For example. But, however, if the fir st card draWn is not replaced then the second draw is dependent on the first dra w.

Exha ustive number of cases n Probability 'p' of the happening of an event is also known as the probability of success'an d the PfObability 'q' of the non- happening of,tile event as the probability of failure. Limitations of Ciassieal Definition. This definitio n of Classic8I Probability breaks down in the following cases: If a trial is' repeated.

It is asSUl7U! Symbolically, if in n trials an event E happens m times,. In a leap year which consists of days there'are 52 complete YIks and 2 days over. The following are the P,Ossible combinatioqs for these twO 'over'days: Since out of the above 7 possibilities, 2 vii.

Theory of Probability Example 4,2. A bag contains 3 red, 6 white and 7 blue balls, What is the' probab ility that two balls drawn are white and blue? What is the probability of getting 9 cards of the same suit in one ha nd at a game of bridge? One hand in a game of bridge consists of 13 ca rds. SlCn Example Assuming t! Find the chance that i it is a multiple of5 or 7, ii it is a multiple of3 or 7.

Theory of Probability 47 14,21, i. A committee of 4 people is to be appointed from J officers of the pro duction department, 4 officers of the download department, two officers of the s ales department and 1 chartered aCCOi4ntant, Find the probability offoT1ning. It should have at least one from the download department. The chartered accounta nt must be in the committee.

Hence ii P [ Committee has no purcha: P [ Committee has at least one purch. The two le tters Rand E can occupy llp2 , i. The number of way s in which there will be exactly 4 letters between Rand E are enumerated below: IPPI', in whi ch 4 are of one kind viz. Following are the 8 poss ible combinations of 4-S's coming consecutively:

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