# 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

 Change of origin and unit 4 Grouped distribution 10 The Normal Distribution 16 PROBABILITY AND PROBABILITY DISTRIBUTIONS 19 Expected value of a sum or a product of two variates 26 Theorems of Tohebychef and Bernoulli 32 Cumulative function of a distribution 39 SOME STANDARD DISTRIBUTIONS 46
 Standard errors in moments about a fixed value 133 Covariance of moments of different orders about a fixed value 135 Standard errors of the variance and the standard deviation of a large sample 136 Comparison of the standard deviations of two large samples 137 Sampling covariance of the means of the variables pcu re 140 Relation between the two functions 147 Beta distribution of the first kind 168 153 Linear constraints Degrees of freedom 166

 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 the regression equation 74 Change of units 75 Numerical illustration 76 Correlation of ranks 79 Bivariate probability distributions 80 Variance of a sum of variates 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 Index of correlation 102 Some related regressions 104 Examples V 105 Chapter VI 109 Large samples Test of significance 111 Comparison of large samples 112 Poissonian and Lexian sampling Samples of varying size 114 Sampling of Values of a Variable 48 Random and simple sampling 116 Sampling distributions Standard errors 117 Sampling distribution of the mean 119 Normal population Fiducial limits for unknown mean 121 Comparison of the means of two large samples 122 Standard error of a partition value 124 Examples VI 126 Chapter VII 130 Standard errors of class frequencies 131 Covariance of the frequencies in different classes 132
 Test of goodness of fit 173 Distribution of regression coefficients and correlation ratios 179 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 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 oriterion of classification 210 Calculation of the sums of squares 212 Two criteria of classification 214 The Latin square Three oriteria of classification 217 Significance 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 MULTIVARIATE DISTRIBUTIONS PARTIAL AND MULTIPLE CORRELATIONS 100 Introductory Yules notation 242 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 Reference 263 Index 273 Table of I 188 277