## A First Course Mathematical StatisticsThis 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 |

273 | |

277 | |

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### Common terms and phrases

Bernoulli's theorem Beta variate binomial distribution calculated chapter class frequencies coefficient of correlation column Consequently constant correlation coefficient correlation ratio corresponding covariance cumulative function curve deduce denote difference distributed like x2 dxdy equal Example expected value expressed Fisher formula Gamma distribution given Hence hypothesis independent Gamma variates independent variates integral interval dx large samples line of regression mean square deviation mean value moment generating function moments normal variates normally distributed number of successes obtained pairs of values Poissonian probability density probability differential proved R. A. Fisher random sample range regression equation respectively result Rietz sampling distribution Show Similarly simple sample square deviation standard deviations statistic statistically independent sum of squares taking expected values tends to infinity theorem tion trials unbiased estimate value of x2 values xi variables variate with parameter vertical array virtue weighted means zero