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# PROBABILITY AND STATISTICS S CHAND PDF

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PDF | Linear correlation coefficient Linear regression; Non-linear Text Book: Probability and Statistics By T K V Iyengar S chand, 3 rd Edition. PROBABILITY AND STATISTICS: (FOR 2ND YEAR pixia-club.info STUDENTS OF JNTU, ANANTAPUR). Dr. T K V Iyengar, Dr. M.V.S.S.N. PRASAD, S. Aim: To acquire basic knowledge in concepts of probability and statistics. Objective: The Edition, Sultan Chand & Sons educational Publishers. 2. Dr. B.S.

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damentals of probability and statistics using mostly calculus. I have given . For an event A of a discrete sample space S, the probability of A can be computed. You may go to the website Library Genesis and search for the book. If not found, search by the name of the book. You will find a list of books relevant to the topic. Fundamentals of Mathematical Statistics By S.C. Gupta – PDF Free Download Publisher: Sultan Chand & Sons; Language: English; ISBN

Are these 'methods correct? Monthly wages in Rs. Age in years: C's method: Each one did his own simplification. B's method: Divide the set into 2. Madurai Univ.. If several sets of observations are combined into a single set. The following data relate to the ages of a group of workers in a factory. Prove the formulae you use. Draw the Ogives and hence estimate the median. Class 32 13 Frequency 8 Defme the weighted arithmetic l.

Show that it is unaffeci. The following table shows some data collected for the regiQlls of a country: Find the values of x and y. Wages Rs. Find the mean of the marks obtained by both the groups of studel ts taken together. What is the mean of the remaining students? Assuming that frequencies are uniformly distributed over the entire interval. Find the mean of the combined distribution. Find the number of boys and number of girls in the class.

The mean weight of boys in the class is 70 kilograms and that of the girls is 55 kilograms. From the following data. A number of particular articles has been classified according to their weights. Substituting in. Statistics Using Median fonnula. Since a and b. Some frequencies a and b in the first weighing and x and y in the second are missing. 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.

Find out the values of the missing frequencies. Now the median of 2nd weighing gives: Find the median. Age of head offamily. Check your results by the graphical method.

Weekly wages No. Upper and lower quartiles. Marks No. Point out the method of finding out the values o median. What is your speed for the entire distance? The salesman replied that the average is only Rs. The following table shows the distribution of families according to their expenditure per week..

On the return trip he makes 15 miles per gallon. A large part of the distance is uphill and he gets a mileage of only 10 per gallon of gasoline. You are required to: Find the harmonic mean of his mileage. The followiflg data represent travel expenses other than transportation for 7 trips made during November by a salesman for a small fmn: Trip Days Expense Expense per day Rs. What is the average speed in m. Number ot families corresponding to expenditure groups Rs.

You take a trip which entails travelling miles by train at an average speed of 60 m. The weights for the fll"st and the second numbers are 2 and 4 respectively. Correct value of the geometic mean. Find the weight of the third. G and H are respectively the arithmetic mean.

G' is given by G.. A variate takes the 'values a.. Expenditure 2Q No. Expenditure No. Apply appropriate correction and calculate the correct geometric mean. Show that the geometric. Tn a frequency table. II prove that 5 Use the method indicated above to find the mean of the following distribution: Show that the mean. If XIV is the weighted mean of x.

Designation Monthly salary in Rs. This is true of all data. Latter is more representative.. Treating the number of letters in each word in the follow: It is a waste of time to apply the refined theoretical methods of Statistics to data which are suspect from the beginning. The sum of squares of deviations is minimum when taken from arithmetic mean and the sum of absolute deviations is minimum when taken from median.

Find the minimum value of: When does the equality sign hold? Match the correct parts to make a valid.. G and H be the arithmetic mean. Fill in blanks: Which of the foltowing are true in respect of any distribution'? Which of the following are true for all sets of data'?

Find out the missing figures: It was later found that it is actually The value of the largest item is Which measure of location win be. Then the median strength is.

Vii Whenxi and Yi are two variab: For the questions given below. The mean of a combined set is given by '" a If wrong observations are replaced: U For the wholesale manufactUrers interested in the type which is usually in demand.

So for working days total attendance is 3. Fundamentals or MadJematical Statistics. In case. So the average speed for the whole journey is either 35 m. I The harmonic mean of n numbers is the reciprocal of the Arithmetic mean of the reCiprocals of the numbers. The average rate 'per rupee is lkg. Do you agree? Hence the production has ". Be brief in your answer: Tendency 2'43 XX In calculating the mean for grouped data. Ue value of mode cannot be calculated exactly. Which average. He drove on the first day at the rate ofA5 Ion per hour.

It further revealed that the grandfathers of these children were also highly intelligent. Is the conclusion right? Hence export trade is not vital to the economy of that country. It was found later th? Hence the teaching standard in college X was hener. In the same year and at the same examination only out of students were successful in college Y. Their correct salaries were and 'Rs. Find the correct arithmetic mean. Hence it i's safe to join military serviC'e than to live in the city of Hyderabad.

So the workers were more efficient in January. Later on. Is the infonnation given above'correct? If not. Skewness and Kurtosis 3. If we know the average alone we cannot forlll a complete idea about tbe distribution as will be cear from the following example. One such measure is Dispersion.

In all tbcse cases we sce that n. Measures of Dispersion. If we arc givcn that the mea n of 5 observations is 9. In the above case we say that series i is more homogeneous less dispersed tban the series ii Qr iii or we say that series iii is 1. The dcsiderdta. Jllust be supported and supplemented by some other measures.

Jiablc mcasure of. Tbe range is the. We study dispersion to bave aJl idea ahout the bQlllpgeneit.. Measures 'of Dispersion.

Range is tbe simples I but a crude lIIeasure of dispersion. Consider the scries i 7. IIore heterogeneous Illore s 'anert"d thall the series i or ii. Quartile deviation or semi-inlcrqulIrtile rangl'. The step of squaring the deviations Xi. But the step of ignoring the signs of the deviations tr. Since mean devilltion is based on all the observations. Ixi -A I I. A creates artificiality and. Xi-X'P I. For the frequency distribution Xi If.

Taking into COnSideration the pros and cons and also the wide applications of standard deviation in statistical theory. The square of standard deviation is called the variance and is given by. It lIlay also be pointed out that standard deviittion gives greater weight to extreme values and as such has not found favour with economists or businessmen who are more interested in Hie results of the modal class'.

X overcomes the drawback of ignoring the signs in mean deviation.. Moreover of all the measures.. The same result could be obtained altematively as follows: Mean square deviation is given by. Different J'ormulae For Calculating Variance.

Thus s. By definition. Measures or Dispersion. A' Obviously s: We have 0'. By definition.. But 'J.. Relation between 0 and s.. Generally the poin. If the values ofx and f are larg. In order to overcome this difficulty.

S out to be in fractions. In that. If we take d. Then We have to prove that.. We know that the mean of a series in A. Find the mean deviation from tire mean ant! Hence the result. Prove that for any discrete distribution standard deviation is not less than mean deviation from mean. Let Xi If. Let Xi If.: Skewness and Kurtosis e r. Ii Xi. Hence Neglecting x. Also l: We bave l. Mneglecting higher powers. For a group of candidates.

Find the corrected mean and standard deviation corresponding to tlte corrected figures. Measures of Dispersio! Corrected mean.. Skewness and.. IM 3 and bigber powers of x. Let then XI i. Ie mean oJ t Ie comb' med series. X2 the means. Variance of the combined series. Skewness and Kurtosis I x2i. Substituting from 3' lOb and 3'10c in 3' Xi and OJ. We have. A -'8 1. The series having greater C. Whenever we want to compare the variability of tht WO series which differ widely in their averages or which are measured in different units.

Co-efficient of Variation. Based upon mean deviation: Average from whkh it is calculated Based'upon standard deviation: Co-efficient of Dispersion. Based upon qua nile deviation: For comparing the variability 01 two series. IX6 Total waces paid AH: Sinl'e C. Iii a The average monthly wages l.

Measures' of Dlspersiob. OI O No. OQ FirmA Firm B N.

In u'IIiill Jirm. The combined variancc 0: Clllicut Unlv. Prove that the mean deviation about the mean frequency of whose ith size Xj is It is given by Number 9 16 12 26 14 12 6 5 x of the variate x. April b Compute quartile deviatIOn graphically for the following data: Marks Classes Frequency Less than 20 30 20 Jive and absolute measures of dispersion and describe the merits and demerits of standard deviation.

Baroda U. Calculate the mean and standard deviation of the following distribution: Of the 10 figures. Explain clearly the ideas impJied in using arbitrary working orgin. The arithmetic mean and variance of a set of 10 figures are known to be 17 and 33 respectively. Standard Deviation 0. What is standard deviation? Explain its superiority over other measures of dispersion.

OJ the wrong item is omitted. Calculate the mean and standard deviation if '. Mean dev.

Golfer B is more. It three of the five observation are 1. Golfer A: The sum and 'sum of squares corresponding to length X in' ems. In which firm A or B. Calculate the. Team B: Firm A Firm 1J No. Which model shows more uniformity? Standard deviation. Z 1 i Which firm. Of all tJie workers 'in the two firms. A and B belonging to the same 'industry. Combined mean. A collar manufacturer is considering the production of a new style collar to attract young men.

Obtain the mean and standa: Obtain the mean and standard deviation of the sample of size obtained by combining the two samples. In a certain test for which the pass marks is Combined S. Nagpur Univ. If i is the mean value of all the measurements. In a series of measuremen The corresponding figures for girls including the 10 failed were 35 and 9. If the deviations are small compared with the value of the mean. Find the mean deviation and variance..

Deihl Unlv. If the mean and standard deviation of a variable x m and CJ respectively. Wiothout quetioning the propriety of this argument. In a frequency distribution. I Ina. Find the mean and variance of first n-natural numbers. Measures of Dispersion.. G2 II M. M is very small in comparison with mean M and X. Deihi Univ. G M where G is the geometric JAean of the v I MecI. G2 t '3"! Froln' a. I'M '2 and hig. Se Stat The rth moment of a variable about the rt1ean given by I 1. We know that if d.!.

Relation between"moments about mean in terms of moments about any point and vice verso. Sheppard's Corrections for Moments.. In particular.

I I uY - 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 proved that if i the frequency distribution is continuous. But since the assumption is not in general true.

Skewness and Kurtosis wherex.

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In case of grouped frequency distribution. Karl Pearson defined the following four coefficients. Factorial moment of order r about the origin of the frequency distribution X..

Alpha ex coefficient is used.. Factorial Moments.

Alpha coefficients are defined as: The following identitfes 'ift.. Charlier's Checks. Moments about mean: Ror the frequency distribution XI if. In the usual notations. The 11h absolute moment of the variable about the 1lean -N'I. PI lind f3!. Absolute Moments. Moments about the point x 2. Comment upon tlte nature 'of distribution. Obtain tlte first four moments abom the orgin.

J4 Comments of. Nature of distribution: O Dividing throughout by N and using reiatiOll 1. If for a random variable x. Establish the relationship between the moments about mean. Obtain as far as possible. The first three moments of a distribution about the value 2 of the variable arc I. II and 15 respectively.. The first four moments of distribution. What are Sheppard's corrections to the central moments?

Find the first four moments about tlie origin. What is the effect of change of origin and s. Obtain mean. Show that the mean is 3. The first four moments of a distribution about.

If Xi If. Mean approx. III 0 because distribution is symmetrical. Delhi Univ. What must be the value of the fourth moment about the mean in order that the distribution be i leptokurtic. If It is the magnitude of the class interval.

Calculate the l'orrect frequency constants. Correct Mean:. It was. In calculating tbe moments of a frequency distribution based on observations. CC of correct values 62 al. Required adjustement. OiJ1ainthe correct va lue of the first Jour centra I moments. These are the absolute measures of. M 0 moderately asymmetrical distribution. BangaIore Uhiv. X moments of aU order exist.

As in dispersion. Measures of Skewness. Mtf - Md. If mode is iII-defined. LimitsIor Bowley's Coefficieitt of Skewness.. I When the.. Clqss i. I cannot even describe how much Course Hero helped me this summer.

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