# A First Course Mathematical Statistics

CUP Archive, Jan 2, 1949 - Mathematics - 271 pages
This book provides the mathematical foundations of statistics. Its aim is to explain the principles, to prove the formulae to give validity to the methods employed in the interpretation of statistical data. Many examples are included but, since the primary emphasis is on the underlying theory, it is of interest to students of a wide variety of subjects: biology, psychology, agriculture, economics, physics, chemistry, and (of course) mathematics.

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### Contents

 Chapter I 1 Change of origin and unit 4 Variance Standard deviation 6 Moments 8 Grouped distribution 10 Continuous distributions 12 EXAMPLES I 16 PROBABILITY AND PROBABILITY DISTRIBUTIONS 7 Explanation of terms Measure of probability 19
 Gamma distribution and Gamma variates 149 Sum of independent Gamma variates 151 Beta distribution of the first kind 153 Alternative proof of theorems 154 Produot of a Bill m variate and a yl+m variate 156 Quotient of independent Gamma variates 158 EXAMPLES VIII 160 Chapter IX 164

 Theorems of total and compound probability 21 Probability distributions Expected valuo 24 Expected value of a sum or a product of two variates 26 Repeated trials Binomial distribution 28 Continuous probability distributions 30 Theorems of Tchebychef and Bornoulli 32 Empirical definition of probability 34 Moment generating function and characteristic function 36 Cumulative function of a distribution 39 EXAMPLES II 42 Chapter III 46 Poissons distribution 47 Derivation from the binomial distribution page 50 Some properties of the normal distribution 51 Probabilities and relative frequencies for various intervals 55 Distribution of a sum of independent normal variates 57 EXAMPLES III 58 MATHEMATICAL NOTES 63 Chapter IV 67 Continuous distributions 68 Lines of regression 69 Coefficient of correlation Standard error of estimate 72 Estimates from tho regression equation 74 Change of units 75 Numerical illustration 76 Correlation of ranks 79 Bivariate probability distributions 80 Variance of a sum of variatos 82 EXAMPLES IV 83 FURTHER CORRELATION THEORY CURVED REGRESSION LINES 33 Arrays Linear regression 87 Correlation ratios 89 Calculation of correlation ratios 91 Other relations 92 Continuous distributions 93 Bivariate normal distribution 95 Intraclass correlation 97 Polynomial regression Normal equations page 99 Large samples Test of significance 111 Sampling distributions Standard orrors 117 Standard error of a partition value 124 Standard errors of class frequencies 131 Comparison of the standard deviations of two large samples 137 Sampling from a Bivariate Population 61 Sampling covariance of the means of the variables page 138 Variance and covariance of moments about a fixed point 140 Standard error of the covariance of a large sample 141 EXAMPLES VII 144 Chapter VIII 146 Relation between the two functions 147
 Linear constraints Degrees of freedom 166 population 169 Nature of the chisquare test An illustration 170 Test of goodness of fit 173 Numerical examples 175 Additive property of chisquare 177 Distribution of regression coefficients and correlation ratios 179 EXAMPLES IX 181 FURTHER TESTS OF SIGNIFICANCE SMALL SAMPLES 84 Small samples page 185 Students Distribution 85 The statistic t and its distribution 186 Test for an assumed population mean 189 Comparison of the means of two samples 190 Significance of an observed correlation 192 Significance of an observed regression coefficient 194 Distribution of the range of a sample 195 Distribution of the Variance Ratio 92 Ratio of independent estimates of the population variance 196 Fishers z distribution Table of F 198 Fishers Transformation of the Correlation Coefficient 94 Distribution of r Fishers transformation 200 Comparison of correlations in independent samples 202 Combination of estimates of a correlation coefficient 203 EXAMPLES X 205 Chapter XI 209 Homogeneous population One criterion of classification 210 Calculation of tho sums of squares 212 Two oriteria of classification 214 The Latin square Three oritoria of classification 217 Significanco of an observed correlation ratio 221 Significance of a regression funotion 223 Test for nonlinearity of regression 224 Resolution of the sum of products One criterion of classification page 226 Calculation of the sums of products 228 Examination and elimination of the effect of regression 229 Two criteria of classification 233 EXAMPLES XI 236 Chapter XII 242 Introductory Yules notation 242 109 Introductory Yules notation 110 Distribution of three or more variables 244 Determination of the coefficients of regression 246 Multiple correlation 249 Partial correlation 250 Reduction formula for the order of a standard deviation 253 Reduction formula for the order of a regression coefficient 254 Normal distribution 255 Significance of an observed partial correlation 256 Significance of an observed multiple correlation 257 EXAMPLES XII 260 LITERATURE FOR REFERENA 263 INDEX 273

### References to this book

 The Art of Smooth PastingAvinash K. DixitLimited preview - 1993
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