## 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

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 |

273 | |

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### Common terms and phrases

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